LUMINOUS
Greg Egan

Perhaps the hottest and fastest-rising new writer to debut in SF in the nineties, Australian Greg Egan is poised on the verge of being recognized as one of the genre's Big Names. In the last few years, Egan has become a frequent contributor to Interzone and Asimov's Science Fiction, and has made sales as well to Pulphouse, Analog, Aurealis, Eidolon, and elsewhere. Several of his stories have appeared in various best-of-the-year series, including this one; in fact, he placed two stories in both our Eighth and Ninth Annual Collections—the first author ever to do that back-to-back in consecutive volumes. He has also had stories in our Tenth, Eleventh, and Twelfth Annual Collections as well. He was on the Hugo Final Ballot in 1995 for his story "Cocoon," which won the Ditmar Award and the Asimov's Readers Award. His first novel, Quarantine, appeared in 1992, to wide critical acclaim, and was followed by a second novel in 1994, Permutation City, which won the John W. Campbell Memorial Award. His most recent book is a collection of his short fiction Axiomatic. Upcoming are two new novels, Distress and Diaspora.
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 triceps 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 couldn't 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 hand-held 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 bum 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 cheerfully, "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 ;r!" 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, and 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 truth 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 space-time 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 comer 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 В and С—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 space-time 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 С—which the quarks have also made true—to the contradiction, the dreaded not-D . . . does it succeed, or doesn't it?"
Alison side-stepped 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 non-standard 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 modem 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—?" Alison sighed theatrically. "I know, I know. And this would bore them witless, poor things—so the subject's closed, I promise." She added wickedly, "Humanities students are so myopic."
We set off across the tranquil leafy campus. Alison kept her word, and we walked in silence; carrying on the argument up to the last minute would have made it even harder to avoid the topic once we were in polite company.
Half-way to the cafeteria, though, I couldn't help myself.
"If someone ever did program a computer to follow a chain of inferences across the defect . . . what do you claim would actually happen? When the end result of all those simple, trustworthy logical steps finally popped up on the screen—which group of primordial quarks would win the battle? And please don't tell me that the whole computer just conveniently vanishes."
Alison smiled, tongue-in-cheek at last. "Get real, Bruno. How can you expect me to answer that, when the mathematics needed to predict the result doesn't even exist yet? Nothing I could say would be true or false—until someone's gone ahead and done the experiment."
I spent most of the day trying to convince myself that I wasn't being followed by some accomplice (or rival) of the surgeon, who might have been lurking outside the hotel. There was something disturbingly Kafkaesque about trying to lose a tail who might or might not have been real: no particular face I could search for in the crowd, just the abstract idea of a pursuer. It was too late to think about plastic surgery to make me look Han Chinese—Alison had raised this as a serious suggestion, back in Vietnam—but Shanghai had over a million foreign residents, so with care even an Anglophone of Italian descent should have been able to vanish.
Whether or not I was up to the task was another matter.
I tried joining the ant-trails of the tourists, following the path of least resistance from the insane crush of the Yuyuan Bazaar (where racks bursting with ten-cent watch-PCs, mood-sensitive contact lenses, and the latest karaoke vocal implants, sat beside bamboo cages of live ducks and pigeons) to the one-time residence of Sun Yatsen (whose personality cult was currently undergoing a mini-series-led revival on Star TV, advertised on ten thousand buses and ten times as many T-shirts). From the tomb of the writer Lu Xun ("Always think and study . . . visit the general then visit the victims, see the realities of your time with open eyes"— no prime time for him) to the Hongkou McDonald's (where they were giving away small plastic Andy Warhol figurines, for reasons I couldn't fathom). I mimed leisurely window-shopping between the shrines, but kept my body language sufficiently unfriendly to deter even the loneliest Westerner from attempting to strike up a conversation. If foreigners were unremarkable in most of the city, they were positively eye-glazing here—even to each other—and I did my best to offer no one the slightest reason to remember me.
Along the way I checked for messages from Alison, but there were none. I left five of my own, tiny abstract chalk marks on bus shelters and park benches—all slightly different, but all saying the same thing: CLOSE BRUSH, BUT SAFE NOW. MOVING ON.
By early evening, I'd done all I could to throw off my hypothetical shadow, so I headed for the next hotel on our agreed but unwritten list. The last time we'd met face-to-face, in Hanoi, I'd mocked all of Alison's elaborate preparations. Now I was beginning to wish that I'd begged her to extend our secret language to cover more extreme contingencies. FATALLY WOUNDED. BETRAYED YOU UNDER TORTURE. REALITY DECAYING. OTHERWISE FINE.
The hotel on Huaihai Zhonglu was a step up from the last one, but not quite classy enough to refuse payment in cash. The desk clerk made polite small-talk, and I lied as smoothly as I could about my plans to spend a week sight-seeing before heading for Beijing. The bellperson smirked when I tipped him too much— and I sat on my bed for five minutes afterward, wondering what significance to read into that.
I struggled to regain a sense of proportion. Industrial Algebra could have bribed every single hotel employee in Shanghai to be on the lockout for us—but that was a bit like saying that, in theory, they could have duplicated our entire twelve-year search for defects, and not bothered to pursue us at all. There was no question that they wanted what we had, badly—but what could they actually do about it? Go to a merchant bank (or the Mafia, or a Triad) for finance? That might have worked if the cargo had been a stray kilogram ofplutonium, or a valuable gene sequence— but only a few hundred thousand people on the planet would be capable of understanding what the defect was, even in theory. Only a fraction of that number would believe that such a thing could really exist . . . and even fewer would be both wealthy and immoral enough to invest in the business of exploiting it.
The stakes appeared to be infinitely high—but that didn't make the players omnipotent. Not yet.
I changed the dressing on my arm, from sock to handkerchief, but the incision was deeper than I'd realized, and it was still bleeding thinly. I left the hotel—and found exactly what I needed in a twenty-four-hour emporium just ten minutes away. Surgical grade tissue repair cream: a mixture of collagen-based adhesive, antiseptic, and growth factors. The emporium wasn't even a pharmaceuticals outlet—it just had aisle after aisle packed with all kinds of unrelated odds and ends, laid out beneath the unblinking blue-white ceiling panels. Canned food, PVC plumbing fixtures, traditional medicines, rat contraceptives, video ROMS. It was a random cornucopia, an almost organic diversity—as if the products had all just grown on the shelves from whatever spores the wind had happened to blow in.
I headed back to the hotel, pushing my way through the relentless crowds, half seduced and half sickened by the odors of cooking, dazed by the endless vista of holograms and neon in a language I barely understood. Fifteen minutes later, reeling from the noise and humidity, I realized that I was lost.
I stopped on a street comer and tried to get my bearings. Shanghai stretched out around me, dense and lavish, sensual and ruthless—a Darwinian economic simulation self-organized to the brink of catastrophe. The Amazon of commerce: this city of sixteen million had more industry of every kind, more exporters and importers, more wholesalers and retailers, traders and re-sellers and re-cyclers and scavengers, more billionaires and more beggars, than most nations on the planet. Not to mention more computing power.
China itself was reaching the cusp of its decades-long transition from brutal totalitarian communism to brutal totalitarian capitalism: a slow seamless morph from Мао to Pinochet set to the enthusiastic applause of its trading partners and the international financial agencies. There'd been no need for a counter-revolution— just layer after layer of carefully reasoned Newspeak to pave the way from previous doctrine to the stunningly obvious conclusion that private property, a thriving middle class, and a few trillion dollars worth of foreign investment were exactly what the Party had been aiming for all along.
The apparatus of the police state remained as essential as ever. Trade unionists with decadent bourgeois ideas about uncompetitive wages, journalists with counter-revolutionary notions of exposing corruption and nepotism, and any number of subversive political activists spreading destabilizing propaganda about the fantasy of free elections, all needed to be kept in check.
In a way. Luminous was a product of this strange transition from communism to not-communism in a thousand tiny steps. No one else, not even the U.S. defense research establishment, possessed a single machine with so much power. The rest of the world had succumbed long ago to networking, giving up their imposing supercomputers with their difficult architecture and customized chips for a few hundred of the latest mass-produced work stations. In fact, the biggest computing feats of the twenty-first century had all been fanned out over the Internet to thousands of volunteers, to run on their machines whenever the processors would otherwise be idle. That was how Alison and I had mapped the defect in the first place: seven thousand amateur mathematicians had shared the joke, for twelve years.
But now the net was the very opposite of what we needed—and only Luminous could take its place. And though only the People's Republic could have paid for it, and only the People's Institute for Advanced Optical Engineering could have built it ... only Shanghai's QIPS Corporation could have sold time on it to the world—while it was still being used to model hydrogen bomb shock waves, pilotless fighter jets, and exotic anti-satellite weapons.
I finally decoded the street signs, and realized what I'd done: I'd turned the wrong way coming out of the emporium, it was as simple as that. I retraced my steps, and I was soon back on familiar territory.
When I opened the door of my room, Alison was sitting on the bed. I said, "What is it with locks in this city?"
We embraced, briefly. We'd been lovers, once—but that was long over. And we'd been friends for years afterward—but I wasn't sure if that was still the right word. Our whole relationship now was too functional, too spartan. Everything revolved around the defect, now.
She said, "I got your message. What happened?" I described the morning's events. "You know what you should have done?" That stung. "I'm still here, aren't I? The cargo's still safe." "You should have killed her, Bruno."
I laughed. Alison gazed back at me placidly, and I looked away. I didn't know if she was serious—and I didn't much want to find out.
She helped me apply the repair cream. My toxin was no threat to her—we'd both installed exactly the same symbionts, the same genotype from the same unique batch in Hanoi. But it was strange to feel her bare fingers on my broken skin, knowing that no one else on the planet could touch me like this, with impunity.
Ditto for sex, but I didn't want to dwell on that.
As I slipped on my jacket, she said, "So guess what we're doing at five a.m. tomorrow?"
"Don't tell me: I fly to Helsinki, and you fly to Cape Town. Just to throw them off the scent."
That got a faint smile. "Wrong. We're meeting Yuen at the Institute—and spending half an hour on Luminous."
"You are brilliant." I bent over and kissed her on the forehead. "But I always knew you'd pull it off."
And I should have been delirious—but the truth was, my guts were churning; I felt almost as trapped as I had upon waking cuffed to the bed. If Luminous had remained beyond our reach (as it should have, since we couldn't afford to hire it for a microsecond at the going rate) we would have had no choice but to destroy all the data, and hope for the best. Industrial Algebra had no doubt dredged up a few thousand fragments of the original Internet calculations—but it was clear that, although they knew exactly what we'd found, they still had no idea where we'd found it. If they'd been forced to start their own random search—constrained by the need for secrecy to their own private hardware—it might have taken them centuries.
There was no question now, though, of backing away and leaving everything to chance. We were going to have to confront the defect in person.
"How much did you have to tell him?"
"Everything." She walked over to the washbasin, removed her shirt, and began wiping the sweat from her neck and torso with a washcloth. "Short of handing over the map. I showed him the search algorithms and their results, and all the programs we'll need to run on Luminous—all stripped of specific parameter values, but enough for him to validate the techniques. He wanted to see direct evidence of the defect, of course, but I held out on that."
"And how much did he believe?"
"He's reserved judgment. The deal is, we get half an hour's unimpeded access— but he gets to observe everything we do."
I nodded, as if my opinion made any difference, as if we had any choice. Yuen Ting-fu had been Alison's supervisor for her Ph.D. on advanced applications of ring theory, when she'd studied at Fu-tan University in the late nineties. Now he was one of the world's leading cryptographers, working as a consultant to the military, the security services, and a dozen international corporations. Alison had once told me that she'd heard he'd found a polynomial-time algorithm for factoring the product of two primes; that had never been officially confirmed ... but such was the power of his reputation that almost everyone on the planet had stopped using the old RSA encryption method as the rumor had spread. No doubt time on Luminous was his for the asking—but that didn't mean he couldn't still be imprisoned for twenty years for giving it away to the wrong people, for the wrong reasons.
I said, "And you trust him? He may not believe in the defect now, but once he's convinced—"
"He'll want exactly what we want. I'm sure of that."
"Okay. But are you sure IA won't be watching, too? If they've worked out why we're here, and they've bribed someone—"
Alison cut me off impatiently. "There are still a few things you can't buy in this city. Spying on a military machine like Luminous would be suicidal. No one would risk it.''
"What about spying on unauthorized projects being run on a military machine? Maybe the crimes cancel out, and you end up a hero."
She approached me, half naked, drying her face on my towel. "We'd better
hope not."
I laughed suddenly. "You know what I like most about Luminous? They're not really letting Exxon and McDonnell-Douglas use the same machine as the People's Liberation Army. Because the whole computer vanishes every time they pull the plug. There's no paradox at all, if you look at it that way."
Alison insisted that we stand guard in shifts. Twenty-four hours earlier, I might have made a joke of it; now I reluctantly accepted the revolver she offered me, and sat watching the door in the neon-tinged darkness while she went out like a
light.
The hotel had been quiet for most of the evening—but now it came to life. There were footsteps in the corridor every five minutes—and rats in the walls, foraging and screwing and probably giving birth. Police sirens wailed in the distance; a couple screamed at each other in the street below. I'd read somewhere that Shanghai was now the murder capital of the world—but was that per capita, or in absolute
numbers?
After an hour, I was so jumpy that it was a miracle I hadn't blown my foot off. I unloaded the gun, then sat playing Russian roulette with the empty barrel. In spite of everything, I still wasn't ready to put a bullet in anyone's brain for the sake of defending the axioms of number theory.
Industrial Algebra had approached us in a perfectly civilized fashion, at first. They were a small but aggressive UK-based company, designing specialized high-performance computing hardware for industrial and military applications. That they'd heard about the search was no great surprise—it had been openly discussed on the Internet for years, and even joked about in serious mathematical journals— but it seemed an odd coincidence when they made contact with us just days after Alison had sent me a private message from Zurich mentioning the latest "promising" result. After half a dozen false alarms—all due to bugs and glitches—we'd stopped broadcasting the news of every unconfirmed find to the people who were donating runtime to the project, let alone any wider circle. We were afraid that if we cried wolf one more time, half our collaborators would get so annoyed that they'd withdraw their support.
IA had offered us a generous slab of computing power on the company's private network—several orders of magnitude more than we received from any other donor. Wiy? The answer kept changing. Their deep respect for pure mathematics . . . their wide-eyed fun-loving attitude to life . . . their desire to be seen to be sponsoring a project so wild and hip and unlikely to succeed that it made SETI look like a staid blue-chip investment. It was—they'd finally "conceded"—a desperate bid to soften their corporate image, after years of bad press for what certain unsavory governments did with their really rather nice smart bombs.
We'd politely declined. They'd offered us highly paid consulting jobs. Bemused, we'd suspended all net-based calculations—and started encrypting our mail with a simple but highly effective algorithm Alison had picked up from Yuen.
Alison had been collating the results of the search on her own work station at her current home in Zurich, while I'd helped coordinate things from Sydney. No doubt IA had been eavesdropping on the incoming data, but they'd clearly started too late to gather the information needed to create their own map; each fragment of the calculations meant little in isolation. But when the work station was stolen (all the files were encrypted, it would have told them nothing) we'd finally been forced to ask ourselves: If the defect turns out to be genuine, if the joke is no joke . . . then exactly what's at stakel How much money^! How much power?
On June 7, 2006, we met in a sweltering, crowded square in Hanoi. Alison wasted no time. She was carrying a backup of the data from the stolen work station in her notepad—and she solemnly proclaimed that, this time, the defect was real.
The notepad's tiny processor would have taken centuries to repeat the long random trawling of the space of arithmetic statements that had been carried out on the net—but, led straight to the relevant computations, it could confirm the existence of the defect in a matter of minutes.
The process began with Statement S. Statement S was an assertion about some ludicrously huge numbers—but it wasn't mathematically sophisticated or contentious in any way. There were no claims here about infinite sets, no propositions concerning "every integer." It merely stated that a certain (elaborate) calculation performed on certain (very large) whole numbers led to a certain result—in essence, it was no different from something like "5 + 3 = 4 x 2".It might have taken me ten years to check it with a pen and paper—but I could have carried out the task with nothing but elementary school mathematics and a great deal of patience. A statement like this could not be undecidable; it had to be either true or false. The notepad decided it was true.
Then the notepad took Statement S . . . and in four hundred and twenty-three simple, impeccably logical steps, used it to prove not-S.
I repeated the calculations on my own notepad—using a different software package. The result was exactly the same. I gazed at the screen, trying to concoct a plausible reason why two different machines running two different programs could have failed in identical ways. There'd certainly been cases in the past of a single misprinted algorithm in a computing textbook spawning a thousand dud programs. But the operations here were too simple, too basic.
Which left only two possibilities. Either conventional arithmetic was intrinsically flawed, and the whole Platonic ideal of the natural numbers was ultimately self-contradictory ... or Alison was right, and an alternative arithmetic had come to hold sway in a "computationally remote" region, billions of years ago.
I was badly shaken—but my first reaction was to try to play down the significance of the result. ' 'The numbers being manipulated here are greater than the volume of the observable universe, measured in cubic Planck lengths. If IA were hoping to use this on their foreign exchange transactions, I think they've made a slight error of scale." Even as I spoke, though, I knew it wasn't that simple. The raw numbers might have been trans-astronomical—but it was the mere 1024 bits of the notepad's binary representations that had actually, physically misbehaved. Every truth in mathematics was encoded, reflected, in countless other forms. If a paradox like this—which at first glance sounded like a dispute about numbers too large to apply even to the most grandiose cosmological discussions—could affect the behavior of a five-gram silicon chip, then there could easily be a billion other systems on the planet at risk of being touched by the very same flaw.
But there was worse to come.
The theory was, we'd located part of the boundary between two incompatible systems of mathematics—both of which were physically true, in their respective domains. Any sequence of deductions that stayed entirely on one side of the defect— whether it was the "near side," where conventional arithmetic applied, or the "far side," where the alternative took over—would be free from contradictions. But any sequence that crossed the border would give rise to absurdities—hence S could lead to not-S.
So, by examining a large number of chains of inference, some of which turned out to be self-contradictory and some not, it should have been possible to map the area around the defect precisely—to assign every statement to one system or the other.
Alison displayed the first map she'd made. It portrayed an elaborately crenulated fractal border, rather like the boundary between two microscopic ice crystals—as if the two systems had been diffusing out at random from different starting points, and then collided, blocking each other's way. By now, I was almost prepared to believe that I really was staring at a snapshot of the creation of mathematics—a fossil of primordial attempts to define the difference between truth and falsehood.
Then she produced a second map of the same set of statements, and overlaid the two. The defect, the border, had shifted—advancing in some places, retreating in others.
My blood went cold. "That has got to be a bug in the software."
"It's not."
I inhaled deeply, looking around the square—as if the heedless crowd of tourists and hawkers, shoppers and executives, might offer some simple "human" truth more resilient than mere arithmetic. But all I could think of was 1984: Winston Smith, finally beaten into submission, abandoning every touchstone of reason by conceding that two and two make five.
I said, "Okay. Go on."
"In the early universe, some physical system must have tested out mathematics that was isolated, cut off from all the established results—leaving it free to decide the outcome at random. That's how the defect arose. But by now, all the mathematics in this region has been tested, all the gaps have been filled in. When a physical system tests a theorem on the near side, not only has it been tested a billion times before— but all the logically adjacent statements around it have been decided, and they imply the correct result in a single step."
"You mean . . . peer pressure from the neighbors? No inconsistencies allowed, you have to conform? If x - 1 = у - 1, and х + 1 = у + 1, then x is left with no choice but to equal у ... because there's nothing 'nearby' to support the alternative?"
"Exactly. Truth is determined locally. And it's the same, deep into the far side. The alternative mathematics has dominated there, and every test takes place surrounded by established theorems that reinforce each other, and the 'correct'— non-standard—result.''
"At the border, though—"
"At the border, every theorem you test is getting contradictory advice. From one neighbor, x - 1 = у - 1 ... but from another, x + 1 = у + 2. And the topology of the border is so complex that a near-side theorem can have more far-side neighbors than near-side ones—and vice versa.
"So the truth at the border isn't fixed, even now. Both regions can still advance or retreat—it all depends on the order in which the theorems are tested. If a solidly near-side theorem is tested first, and it lends support to a more vulnerable neighbor, that can guarantee that they both stay near-side." She ran a brief animation that demonstrated the effect. "But if the order is reversed, the weaker one will fall."
I watched, light-headed. Obscure—but supposedly eternal—truths were tumbling like chess pieces. "And . . . you think that physical processes going on right now— chance molecular events that keep inadvertently testing and re-testing different theories along the border—cause each side to gain and lose territory?"
"Yes."
"So there's been a kind of ... random tide washing back and forth between the two kinds of mathematics, for the past few billion years?" I laughed uneasily, and did some rough calculations in my head. "The expectation value for a random walk is the square root of N. I don't think we have anything to worry about. The tide isn't going to wash over any useful arithmetic in the lifetime of the universe."
Alison smiled humorlessly, and held up the notepad again. "The tide? No. But it's the easiest thing in the world to dig a channel. To bias the random flow." She ran an animation of a sequence of tests that forced the far-side system to retreat across a small front—exploiting a "beachhead" formed by chance, and then pushing on to undermine a succession of theorems. "Industrial Algebra, though—I imagine— would be more interested in the reverse. Establishing a whole network of narrow channels of non-standard mathematics running deep into the realm of conventional arithmetic—which they could then deploy against theorems with practical consequences."
I fell silent, trying to imagine tendrils of contradictory arithmetic reaching down into the everyday world. No doubt IA would aim for surgical precision—hoping to earn themselves a few billion dollars by corrupting the specific mathematics underlying certain financial transactions. But the ramifications would be impossible to predict—or control. There'd be no way to limit the effect, spatially—they could target certain mathematical truths, but they couldn't confine the change to any one location. A few billion dollars, a few billion neurons, a few billion stars . . . a few billion people. Once the basic rules of counting were undermined, the most solid and distinct objects could be rendered as uncertain as swirls of fog. This was not a power I would have entrusted to a cross between Mother Theresa and Carl Friedrich Gauss.
"So what do we do? Erase the map—and just hope that IA never find the defect for themselves?"
"No." Alison seemed remarkably calm—but then, her own long-cherished philosophy had just been confirmed, not razed to the ground—and she'd had time on the flight from Zurich to think through all the Realmathematik. "There's only one way to be sure that they can never use this. We have to strike first. We have to get hold of enough computing power to map the entire defect. And then we either iron the border flat, so it can't move—if you amputate all the pincers, there can be no pincer movements. Or—better yet, if we can get the resources—we push the border in, from all directions, and shrink the far-side system down to nothing.''
I hesitated. "All we've mapped so far is a tiny fragment of the defect. We don't know how large the far side could be. Except that it can't be small—or the random fluctuations would have swallowed it long ago. And it could go on forever; it could be infinite, for all we know."
Alison gave me a strange look. "You still don't get it, do you, Bruno? You're still thinking like a Platonist. The universe has only been around for fifteen billion years. It hasn't had time to create infinities. The far side can't go on forever— because somewhere beyond the defect, there are theorems that don't belong to any system. Theorems that have never been touched, never been tested, never been rendered true or false.
"And if we have to reach beyond the existing mathematics of the universe in order to surround the far side . . . then that's what we'll do. There's no reason why it shouldn't be possible—just so long as we get there first."
When Alison took my place, at one in the morning, I was certain I wouldn't get any sleep. When she shook me awake three hours later, I still felt like I hadn't.
I used my notepad to send a priming code to the data caches buried in our veins, and then we stood together side-by-side, left-shoulder-to-right-shoulder. The two chips recognized each other's magnetic and electrical signatures, interrogated each other to be sure—and then began radiating lower power microwaves. Alison's notepad picked up the transmission, and merged the two complementary data streams. The result was still heavily encrypted—but after all the precautions we'd taken so far, shifting the map into a hand-held computer felt about as secure as tattooing it onto our foreheads.
A taxi was waiting for us downstairs. The People's Institute for Advanced Optical Engineering was in Minhang, a sprawling technology park some thirty kilometers south of the city center. We rode in silence through the gray predawn light, past the giant ugly tower blocks thrown up by the landlords of the new millennium, riding out the fever as the necrotraps and their cargo dissolved into our blood.
As the taxi turned into an avenue lined with biotech and aerospace companies, Alison said, "If anyone asks, we're Ph.D. students ofYuen's, testing a conjecture in algebraic topology."
"Now you tell me. I don't suppose you have any specific conjecture in mind? What if they ask us to elaborate?''
"On algebraic topology? At five o'clock in the morning?"
The Institute building was unimposing—sprawling black ceramic, three stories high—but there was a five-meter electrified fence, and the entrance was guarded by two armed soldiers. We paid the taxi driver and approached on foot. Yuen had supplied us with visitor's passes—complete with photographs and fingerprints. The names were our own; there was no point indulging in unnecessary deception. If we were caught out, pseudonyms would only make things worse.
The soldiers checked the passes, then led us through an MRI scanner. I forced
myself to breathe calmly as we waited for the results; in theory, the scanner could have picked up our symbionts' foreign proteins, lingering breakdown products from the necrotraps, and a dozen other suspicious trace chemicals. But it all came down to a question of what they were looking for; magnetic resonance spectra for billions of molecules had been catalogued—but no machine could hunt for all of them at once.
One of the soldiers took me aside and asked me to remove my jacket. I fought down a wave of panic—and then struggled not to overcompensate: if I'd had nothing to hide, I would still have been nervous. He prodded the bandage on my upper arm; the surrounding skin was still red and inflamed. "What's this?"
"I had a cyst there. My doctor cut it out, this morning." He eyed me suspiciously, and peeled back the adhesive bandage—with ungloved hands. I couldn't bring myself to look; the repair cream should have sealed the wound completely—at worst there should have been old, dried blood—but I could feel a faint liquid warmth along the line of the incision.
The soldier laughed at my gritted teeth, and waved me away with an expression of distaste. I had no idea what he thought I might have been hiding—but I saw fresh red droplets beading the skin before I closed the bandage.
Yuen Ting-fu was waiting for us in the lobby. He was a slender, fit-looking man in his late sixties, casually dressed in denim. I let Alison do all the talking: apologizing for our lack of punctuality (although we weren't actually late), and thanking him effusively for granting us this precious opportunity to pursue our unworthy research. I stood back and tried to appear suitably deferential. Four soldiers looked on impassively; they didn't seem to find all this groveling excessive. And no doubt I would have been giddy with awe, if I really had been a student granted time here for some run-of-the-mill thesis.
We followed Yuen as he strode briskly through a second checkpoint and scanner (this time, no one stopped us) then down a long corridor with a soft gray vinyl floor. We passed a couple of white-coated technicians, but they barely gave us a second glance. I'd had visions of a pair of obvious foreigners attracting as much attention here as we would have wandering through a military base—but that was absurd. Half the runtime on Luminous was sold to foreign corporations—and because the machine was most definitely not linked to any communications network, commercial users had to come here in person. Just how often Yuen wangled free time for his students—whatever their nationality—was another question, but if he believed it was the best cover for us, I was in no position to argue. I only hoped he'd planted a seamless trail of reassuring lies in the university records and beyond, in case the Institute administration decided to check up on us in any detail.
We stopped in at the operations room, and Yuen chatted with the technicians. Banks offlatscreens covered one wall, displaying status histograms and engineering schematics. It looked like the control center for a small particle accelerator—which wasn't far from the truth.
Luminous was, literally, a computer made of light. It came into existence when a vacuum chamber, a cube five meters wide, was filled with an elaborate standing wave created by three vast arrays of high-powered lasers. A coherent electron beam was fed into the chamber—and just as a finely machined grating built of solid matter could diffract a beam of light, a sufficiently ordered (and sufficiently intense) configuration of light could diffract a beam of matter.
The electrons were redirected from layer to layer of the light cube, recombining and interfering at each stage, every change in their phase and intensity performing an appropriate computation—and the whole system could be reconfigured, nanosecond by nanosecond, into complex new "hardware" optimized for the calculations at hand. The auxilliary supercomputers controlling the laser arrays could design, and then instantly build, the perfect machine of light to carry out each particular stage of any program.
It was, of course, fiendishly difficult technology, incredibly expensive and temperamental. The chance of ever putting it on the desktops of Tetris-playing accountants was zero, so nobody in the West had bothered to pursue it.
And this cumbersome, unwieldy, impractical machine ran faster than every piece of silicon hanging off the Internet, combined.
We continued on to the programming room. At first glance, it might have been the computing center in a small primary school, with half a dozen perfectly ordinary work stations sitting on white formica tables. They just happened to be the only six in the world that were hooked up to Luminous.
We were alone with Yuen now—and Alison cut the protocol and just glanced briefly in his direction for approval, before hurriedly linking her notepad to one of the work stations and uploading the encrypted map. As she typed in the instructions to decode the file, all the images running through my head of what would have happened if I'd poisoned the soldier at the gate receded into insignificance. We now had half an hour to banish the defect—and we still had no idea how far it extended.
Yuen turned to me; the tension on his face betrayed his own anxieties, but he mused philosophically, "If our arithmetic seems to fail for these large numbers— does it mean the mathematics, the ideal, is really flawed and mutable—or only that the behavior of matter always falls short of the ideal?"
I replied, "If every class of physical objects 'falls short' in exactly the same way—whether it's boulders or electrons or abacus beads . . . what is it that their common behavior is obeying—or defining—if not the mathematics?"
He smiled, puzzled. "Alison seemed to think you were a Platonist."
"Lapsed. Or ... defeated. I don't see what it can mean to talk about standard number theory still being true for these statements—in some vague Platonic sense— if no real objects can ever reflect that truth."
"We can still imagine it. We can still contemplate the abstraction. It's only the physical act of validation that must fall through. Think of transfinite arithmetic: no one can physically test the properties of Cantor's infinities, can they? We can only reason about them from afar.''
I didn't reply. Since the revelations in Hanoi, I'd pretty much lost faith in my power to "reason from afar" about anything I couldn't personally describe with Arabic numerals on a single sheet of paper. Maybe Alison's idea of "local truth" was the most we could hope for; anything more ambitious was beginning to seem like the comic-book "physics" of swinging a rigid beam ten billion kilometers long around your head, and predicting that the far end would exceed the speed of light.
An image blossomed on the work station screen: it began as the familiar map of the defect—but Luminous was already extending it at a mind-boggling rate. Billions of inferential loops were being spun around the margins: some confirming their own premises, and thus delineating regions where a single, consistent mathematics held sway . . . others skewing into self-contradiction, betraying a border crossing. I tried to imagine what it would have been like to follow one of those Mobius-strips of deductive logic in my head; there were no difficult concepts involved, it was only the sheer size of the statements that made that impossible. But would the contradictions have driven me into gibbering insanity—or would I have found every step perfectly reasonable, and the conclusion simply unavoidable? Would I have ended up calmly, happily conceding: Two and two makefive7
As the map grew—smoothly re-scaled to keep it fitting on the screen, giving the unsettling impression that we were retreating from the alien mathematics as fast as we could, and only just avoiding being swallowed—Alison sat hunched forward, waiting for the big picture to be revealed. The map portrayed the network of statements as an intricate lattice in three dimensions (a crude representational convention, but it was as good as any other). So far, the border between the regions showed no sign of overall curvature—just variously sized random incursions in both directions. For all we knew, it was possible that the far-side mathematics enclosed the near side completely—that the arithmetic we'd once believed stretched out to infinity was really no more than a tiny island in an ocean of contradictory truths.
I glanced at Yuen; he was watching the screen with undisguised pain. He said, "I read your software, and I thought: sure, this looks fine—but some glitch on your machines is the real explanation. Luminous will soon put you right." Alison broke in jubilantly, "Look, it's turning!"
She was right. As the scale continued to shrink, the random fractal meanderings of the border were finally being subsumed by an overall convexity—a convexity of the far side. It was as if the viewpoint was backing away from a giant spiked sea-urchin. Within minutes, the map showed a crude hemisphere, decorated with elaborate crystalline extrusions at every scale. The sense of observing some paleo-mathematical remnant was stronger than ever, now: this bizarre cluster of theorems really did look as if it had exploded out from some central premise into the vacuum of unclaimed truths, perhaps a billionth of a second after the Big Bang—only to be checked by an encounter with our own mathematics.
The hemisphere slowly extended into a three-quarters sphere . . . and then a spiked whole. The far side was bounded, finite. It was the island, not us.
Alison laughed uneasily. "Was that true before we started—or did we just make it true?" Had the near side enclosed the far side for billions of years—or had Luminous broken new ground, actively extending the near side into mathematical territory that had never been tested by any physical system before.
We'd never know. We'd designed the software to advance the mapping along a ; front in such a way that any unclaimed statements would be instantly recruited into ' the near side. If we'd reached out blindly, far into the void, we might have tested 'i an isolated statement—and inadvertently spawned a whole new alternative mathematics to deal with.
Alison said, "Okay—now we have to decide. Do we try to seal the border—or do we take on the whole structure?" The software, I knew, was busy assessing the relative difficulty of the tasks.
Yuen replied at once, "Seal the border, nothing more. You mustn't destroy this." He turned to me, imploringly. "Would you smash up a fossil of Australopi-thecus'? Would you wipe the cosmic background radiation out of the sky? This may shake the foundations of all my beliefs—but it encodes the truth about our history. We have no right to obliterate it, like vandals."
Alison eyed me nervously. What was this—majority rule7 Yuen was the only one with any power here; he could pull the plug in an instant. And yet it was clear from his demeanor that he wanted a consensus—he wanted our moral support for any decision.
I said cautiously, "If we smooth the border, that'll make it literally impossible for IA to exploit the defect, won't it?"
Alison shook her head. "We don't know that. There may be a quantumlike component of spontaneous defections, even for statements that appear to be in perfect equilibrium.''
Yuen countered, "Then there could be spontaneous defections anywhere—even far from any border. Erasing the whole structure will guarantee nothing."
"It will guarantee that IA won't find it! Maybe pin-point defections do occur, all the time—but the next time they're tested, they'll always revert. They're surrounded by explicit contradictions, they have no chance of getting a foothold. You can't compare a few transient glitches with this . . . armory of counter-mathematics!"
The defect bristled on the screen like a giant caltrop. Alison and Yuen both turned to me expectantly. As I opened my mouth, the work station chimed. The software had examined the alternatives in detail: destroying the entire far side would take Luminous twenty-three minutes and seventeen seconds—about a minute less than the time we had left. Sealing the border would take more than an hour.
I said, "That can't be right."
Alison groaned. "But it is! There's random interference going on at the border from other systems all the time—and doing anything finicky there means coping with that noise, fighting it. Charging ahead and pushing the border inward is different: you can exploit the noise to speed the advance. It's not a question of dealing with a mere surface versus dealing with a whole volume. It's more like . . . trying to carve an island into an absolutely perfect circle, while waves are constantly crashing on the beach—versus bulldozing the whole thing into the ocean."
We had thirty seconds to decide—or we'd be doing neither today. And maybe Yuen had the resources to keep the map safe from IA, while we waited a month or more for another session on Luminous—but I wasn't prepared to live with that kind of uncertainty.
"I say we get rid of the whole thing. Anything less is too dangerous. Future mathematicians will still be able to study the map—and if no one believes that the defect itself ever really existed, that's just too bad. IA is too close. We can't risk it."
Alison had one hand poised above the keyboard. I turned to Yuen; he was staring
at the floor with an anguished expression. He'd let us state our views—but in the end, it was his decision.
He looked up, and spoke sadly but decisively.
"Okay. Doit."
Alison hit the key—with about three seconds to spare. I sagged into my chair, light-headed with relief.
We watched the far side shrinking. The process didn't look quite as crass as bulldozing an island—more like dissolving some quirkily beautiful crystal in acid. Now that the danger was receding before our eyes, though, I was beginning to suffer faint pangs of regret. Our mathematics had coexisted with this strange anomaly for fifteen billion years, and it shamed me to think that within months of its discovery, we'd backed ourselves into a comer where we'd had no choice but to destroy it.
Yuen seemed transfixed by the process. "So are we breaking the laws of physics—or enforcing them?"
Alison said, "Neither. We're merely changing what the laws imply."
He laughed softly. " 'Merely.' For some esoteric set of complex systems, we're rewriting the high-level rules of their behavior. Not including the human brain, I hope."
My skin crawled. "Don't you think that's . . . unlikely?"
"I was joking." He hesitated, then added soberly, "Unlikely for humans—but someone could be relying on this, somewhere. We might be destroying the whole basis of their existence: certainties as fundamental to them as a child's multiplication tables are to us."
Alison could barely conceal her scorn. "This is junk mathematics—a relic of a pointless accident. Any kind of life that evolved from simple to complex forms would have no use for it. Our mathematics works for ... rocks, seeds, animals in the herd, members of the tribe. This only kicks in beyond the number of particles in the universe—"
"Or smaller systems that represent those numbers," I reminded her.
"And you think life somewhere might have a burning need to do nonstandard trans-astronomical arithmetic, in order to survive? I doubt that very much."
We fell silent. Guilt and relief could fight it out later, but no one suggested halting the program. In the end, maybe nothing could outweigh the havoc the defect would have caused if it had ever been harnessed as a weapon—and I was looking forward to composing a long message to Industrial Algebra, informing them of precisely what we'd done to the object of their ambitions.
Alison pointed to a comer of the screen. "What's that?" A narrow dark spike protruded from the shrinking cluster of statements. For a moment I thought it was merely avoiding the near side's assault—but it wasn't. It was slowly, steadily growing longer.
"Could be a bug in the mapping algorithm." I reached for the keyboard and zoomed in on the structure. In close-up, it was several thousand statements wide. At its border, Alison's program could be seen in action, testing statements in an order designed to force tendrils of the near side ever deeper into the interior. This slender extrusion, ringed by contradictory mathematics, should have been corroded out of existence in a fraction of a second. Something was actively countering the assault, though—repairing every trace of damage before it could spread.
' 'If IA have a bug here—'' I turned to Yuen. ' They couldn't take on Luminous directly, so they couldn't stop the whole far side shrinking—but a tiny structure like this . . . what do you think? Could they stabilize it?"
"Perhaps," he conceded. "Four or five hundred top-speed work stations could do it."
Alison was typing frantically on her notepad. She said. "I'm writing a patch to identify any systematic interference—and divert all our resources against it. She brushed her hair out of her eyes. "Look over my shoulder, will you, Bruno. Check me as I go." 
"Okay." I read through what she'd written so far. "You're doing fine. Stay calm." Her hands were trembling.
The spike continued to grow steadily. By the time the patch was ready, the map was re-scaling constantly to fit it on the screen.
Alison triggered the patch. An overlay of electric blue appeared along the spike, flagging the concentration of computing power—and the spike abruptly froze.
I held my breath, waiting for IA to notice what we'd done—and switch their resources elsewhere? If they did, no second spike would appear—they'd never get that far—but the blue marker on the screen would shift to the site where they'd regrouped and tried to make it happen. 
But the blue glow didn't move from the existing spike. And the spike didn't vanish under the weight of Luminous's undivided efforts,
Instead, it began to grow again, slowly.
Yuen looked ill. "This is not Industrial Algebra. There's no computer on the planet—" 
Alison laughed derisively. "What are you saying now? Aliens who need the farside are defending it? Aliens where? Nothing we've done has had time to reach even . . . Jupiter." There was an edge of hysteria in her voice.
"Have you measured how fast the changes propagate? Do you know, for certain, that they can't travel faster than light—with the far-side mathematics undermining the logic of relativity?" 
I said, "Whoever it is, they're not defending all their borders. They're putting everything they've got into the spike." 
"They're aiming at something. A specific target." Yuen reached over Alison's shoulder for the keyboard. "We're shutting this down. Right now."
She turned on him, blocking his way. "Are you crazy? We're almost holding them off! I'll rewrite the program, fine-tune it, get an edge in efficiency—
"No! We stop threatening them, then see how they react. We don't know what harm we're doing—"
He reached for the keyboard again. 
Alison jabbed him in the throat with her elbow, hard. He staggered backward, gasping for breath, then crashed to the floor, bringing a chair down on top of him.
She hissed at me, "Quick—shut him up!"
I hesitated, loyalties fracturing; his idea had sounded perfectly sane to me. But if he started yelling for security— 
I crouched down over him, pushed the chair aside, then clasped my hand over his mouth, forcing his head back with pressure on the lower jaw. We'd have to tie him up—and then try brazenly marching out of the building without him. But he'd be found in a matter of minutes. Even if we made it past the gate, we were screwed.
Y caught his breath and started struggling; I clumsily pinned his arms with my knees. I could hear Alison typing, a ragged staccato; I tried to get a glimpse of the work station screen, but I couldn't turn that far without taking my weight off Yuen.
I said, "Maybe he's right—maybe we should pull back, and see what happens."
If the alterations could propagate faster than light . . . how many distant civilizations might have felt the effects of what we'd done? Our first contact with extraterrestrial life could turn out to be an attempt to obliterate mathematics that they viewed as … what? A precious resource? A sacred relic? An essential component of their entire world view?
The sound of typing stopped abruptly. ' 'Bruno? Do you feel—?''
"What?"
 Silence.
"What?"
Yuen seemed to have given up the fight. I risked turning around.
Alison was hunched forward, her face in her hands. On the screen, the spike had ceased its relentless linear growth—but now an elaborate dendritic structure had blossomed at its tip. I glanced down at Yuen; he seemed dazed, oblivious to my presence. I took my hand from his mouth, warily. He lay there placidly, smiling faintly, eyes scanning something I couldn't see.
I climbed to my feet. I took Alison by the shoulders and shook her gently; her gently; her only response was to press her face harder into her hands. The spike's strange flower was still growing—but it wasn't spreading out into new territory; it was sending narrow shoots back in on itself, crisscrossing the same region again and again with ever finer structures.
Weaving a net? Searching for something?
It hit more with a jolt of clarity more intense than anything I'd felt since childhood. It was like reliving the moment when the whole concept of numbers had finally snapped into place—but with an adult's understanding of everything it opened up, everything it implied. It was a lightning-bolt revelation—but there was no taint of mystical confusion: no opiate haze of euphoria, no pseudo-sexual rush. In the clean-lined logicof the simplest concepts, I saw and understood exactly how the world worked—
—except that it was all wrong, it was all false, it was all impossible.
Quicksand.
Assailed by vertigo, I swept my gaze around the room—counting frantically: Six work stations. Two people. Six chairs. I grouped the work stations: three sets of two, two sets of three. One and five, two and four, four and two, five and one.
I weaved a dozen cross-checks for consistency—for sanity ... but everything added up.
They hadn't stolen the old arithmetic; they'd merely blasted the new one into my head, on top of it.
Whoever had resisted our assault with Luminous had reached down with the spike and rewritten our neural metamathematics—the arithmetic that underlay our own reasoning about arithmetic—enough to let us glimpse what we'd been trying to destroy.
Alison was still uncommunicative, but she was breathing slowly and steadily. Yuen seemed fine, lost in a happy reverie. I relaxed slightly, and began trying to make sense of the flood of far-side arithmetic surging through my brain.
On their own terms, the axioms were . . . trivial, obvious. I could see that they corresponded to elaborate statements about trans-astronomical integers, but performing an exact translation was far beyond me—and thinking about the entities they described in terms of the huge integers they represented was a bit like thinking about pi or the square root of two in terms of the first ten thousand digits of their decimal expansion: it would be missing the point entirely. These alien "numbers"— the basic objects of the alternative arithmetic—had found a way to embed themselves in the integers, and to relate to each other in a simple, elegant way—and if the messy corollaries they implied upon translation contradicted the rules integers were supposed to obey . . . well, only a small, remote patch of obscure truths had been subverted.
Someone touched me on the shoulder. I started—but Yuen was beaming amiably, all arguments and violence forgotten.
He said, "Lightspeed is not violated. All the logic that requires that remains intact." I could only take him at his word; the result would have taken me hours to prove. Maybe the aliens had done a better job on him—or maybe he was just a superior mathematician in either system.
"Then . . . where are they?" At lightspeed, our attack on the far side could not have been felt any further away than Mars—and the strategy used to block the corrosion of the spike would have been impossible with even a few seconds' time lag.
"The atmosphere?"
"You mean—Earth's?"
"Where else? Or maybe the oceans."
I sat down heavily. Maybe it was no stranger than any conceivable alternative, but I still balked at the implications.
Yuen said, "To us, their structure wouldn't look like 'structure' at all. The simplest unit might involve a group of thousands of atoms—representing a trans-astronomical number—not necessarily even bonded together in any conventional way, but breaking the normal consequences of the laws of physics, obeying a different set of high-level rules that arise from the alternative mathematics. People have often mused about the chances of intelligence being coded into long-lived vortices on distant gas giants . . . but these creatures won't be in hurricanes or tornadoes. They'll be drifting in the most innocuous puffs of air—invisible as neutrinos."
"Unstable—"
"Only according to our mathematics. Which does not apply."
Alison broke in suddenly, angrily. "Even if all of this is true—where does it get us? Whether the defect supports a whole invisible ecosystem or not—IA will still find it, and use it, in exactly the same way."
For a moment I was dumbstruck. We were facing the prospect of sharing the planet with an undiscovered civilization—and all she could think about was IA's grubby machinations?
She was absolutely right, though. Long before any of these extravagant fantasies i could be proved or disproved, IA could still do untold harm. 
I said, "Leave the mapping software running—but shut down the shrinker."
She glanced at the screen. "No need. They've overpowered it—or undermined its mathematics." The far side was back to its original size. "Then there's nothing to lose. Shut it down."
She did. No longer under attack, the spike began to reverse its growth. I felt a pang of loss as my limited grasp of the far-side mathematics suddenly evaporated.
I tried to hold on, but it was like clutching at air. 
When the spike had retracted completely, I said, "Now we try doing an Industrial Algebra. We try bringing the defect closer."
We were almost out of time, but it was easy enough—in thirty seconds, we rewrote the shrinking algorithm to function in reverse.
Alison programmed a function key with the commands to revert to the original version—so that if the experiment backfired, one keystroke would throw the full weight of Luminous behind a defense of the near side again.
Yuen and I exchanged nervous glances. I said, "Maybe this wasn't such a good idea."
Alison disagreed. "We need to know how they'll react to this. Better we find out now than leave it to IA." She started the program running.
The sea-urchin began to swell, slowly. I broke out in a sweat. The far-siders hadn't harmed us, so far—but this felt like tugging hard at a door that you really, badly, didn't want to see thrown open.
A technician poked her head into the room and announced cheerfully, "Down for maintenance in two minutes!" Yuen said, "I'm sorry, there's nothing—"
The whole far side turned electric blue. Alison's original patch had detected a systematic intervention.
We zoomed in. Luminous was picking off vulnerable statements of the near side—but something else was repairing the damage.
I let out a strangled noise that might have been a cheer. Alison smiled serenely. She said, "I'm satisfied. IA don't stand a chance."
Yuen mused, "Maybe they have a reason to defend the status quo—maybe they rely on the border itself, as much as the far side."
Alison shut down our reversed shrinker. The blue glow vanished; both sides were leaving the defect alone. And there were a thousand questions we all wanted answered—but the technicians had thrown the master switch, and Luminous itself had ceased to exist.
The sun was breaking through the skyline as we rode back into the city. As we pulled up outside the hotel, Alison started shaking and sobbing. I sat beside her, squeezing her hand. I knew she'd felt the weight of what might have happened, all along, far more than I had.
I paid the driver, and then we stood on the street for a while, silently watching the cyclists go by, trying to imagine how the world would change as it tried to embrace this new contradiction between the exotic and the mundane, the pragmatic and the Platonic, the visible and the invisible.