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 farside 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
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 farside 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
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