Luminous (4 page)

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 inc
onsistencies 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'— nonstandard—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 farside 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 farside 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 nonstandard 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 farside 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