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 palaeomathematical 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 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 Australopithecus? 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 rule? 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 quantum-like 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 caltrap. 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 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. Do it."

  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 corner 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 corner 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 far side 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 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.

  Yuen caught his breath and started struggling; I clumsily pinned his arms with my knees. I could hear Alison typing, a ragged staccato, 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 he 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 structure Weaving a net? Searching for something?

  It hit me with a jolt of clarity more intense than anything I'd felt since childhood. It was like reliving the moment when the whole concept 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 logic of 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 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-LA 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 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.

 

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