"Good morning, Bruno. How is the weather there in Sparseland?”
screen icon for my interlocutor was a three-holed torus tiled with
triangles, endlessly turning itself inside out. The polished tones of
the male synthetic voice I heard conveyed no specific origin, but gave
a sense nonetheless that the speaker’s first language was something
other than English.
I glanced out the window of my
home office, taking in a patch of blue sky and the verdant gardens of a
shady West Ryde cul-de-sac. Sam used “good morning” regardless of the
hour, but it really was just after ten a.m., and the tranquil Sydney
suburb was awash in sunshine and birdsong.
“Perfect,” I replied. “I wish I wasn’t chained to this desk.”
was a long pause, and I wondered if the translator had mangled the
idiom, creating the impression that I had been shackled by ruthless
assailants, who had nonetheless left me with easy access to my instant
messaging program. Then Sam said, “I’m glad you didn’t go for a run
today. I’ve already tried Alison and Yuen, and they were both
unavailable. If I hadn’t been able to get through to you, it might have
been difficult to keep some of my colleagues in check.”
felt a surge of anxiety, mixed with resentment. I refused to wear an
iWatch, to make myself reachable twenty-four hours a day. I was a
mathematician, not an obstetrician. Perhaps I was an amateur diplomat
as well, but even if Alison, Yuen, and I didn’t quite cover the time
zones, it would never be more than a few hours before Sam could get
hold of at least one of us.
“I didn’t realize you were
surrounded by hotheads,” I replied. “What’s the great emergency?” I
hoped the translator would do justice to the sharpness in my voice.
Sam’s colleagues were the ones with all the firepower, all the
resources; they should not have been jumping at shadows. True, we had
once tried to wipe them out, but that had been a perfectly innocent
mistake, more than ten years before.
Sam said, “Someone from your side seems to have jumped the border.”
far as we can see, there’s no trench cutting through it. But a few
hours ago, a cluster of propositions on our side started obeying your
I was stunned. “An isolated cluster? With no derivation leading back to us?”
“None that we could find.”
thought for a while. “Maybe it was a natural event. A brief surge
across the border from the background noise that left a kind of tidal
Sam was dismissive. “The cluster was too
big for that. The probability would be vanishingly small.” Numbers came
through on the data channel; he was right.
I rubbed my
eyelids with my fingertips; I suddenly felt very tired. I’d thought our
old nemesis, Industrial Algebra, had given up the chase long ago. They
had stopped offering bribes and sending mercenaries to harass me, so
I’d assumed they’d finally written off the defect as a hoax or a
mirage, and gone back to their core business of helping the world’s
military kill and maim people in ever more technologically
Maybe this wasn’t IA. Alison and I
had first located the defect—a set of contradictory results in
arithmetic that marked the border between our mathematics and the
version underlying Sam’s world—by means of a vast set of calculations
farmed out over the internet, with thousands of volunteers donating
their computers’ processing power when the machines would otherwise
have been idle. When we’d pulled the plug on that project—keeping our
discovery secret, lest IA find a way to weaponize it—a few participants
had been resentful, and had talked about continuing the search. It
would have been easy enough for them to write their own software,
adapting the same open source framework that Alison and I had used, but
it was difficult to see how they could have gathered enough supporters
without launching some kind of public appeal.
I said, “I can’t offer you an immediate explanation for this. All I can do is promise to investigate.”
“I understand,” Sam replied.
have no clues yourself ?” A decade before, in Shanghai, when Alison,
Yuen, and I had used the supercomputer called Luminous to mount a
sustained attack on the defect, the mathematicians of the far side had
grasped the details of our unwitting assault clearly enough to send a
plume of alternative mathematics back across the border with pinpoint
precision, striking at just the three of us.
“If the cluster had been connected to something, we could have followed
the trail. But in isolation it tells us nothing. That’s why my
colleagues are so anxious.”
“Yeah.” I was still hoping
that the whole thing might turn out to be a glitch—the mathematical
equivalent of a flock of birds with a radar echo that just happened to
look like something more sinister—but the full gravity of the situation
was finally dawning on me.
The inhabitants of the far
side were as peaceable as anyone might reasonably wish their neighbors
to be, but if their mathematical infrastructure came under threat they
faced the real prospect of annihilation. They had defended themselves
from such a threat once before, but because they had been able to trace
it to its source and understand its nature, they had shown great
forbearance. They had not struck their assailants dead, or wiped out
Shanghai, or pulled the ground out from under our universe.
new assault had not been sustained, but nobody knew its origins, or
what it might portend. I believed that our neighbors would do no more
than they had to in order to ensure their survival, but if they were
forced to strike back blindly, they might find themselves with no path
to safety short of turning our world to dust.
time was only two hours behind Sydney, but Yuen’s IM status was still
“unavailable.” I emailed him, along with Alison, though it was the
middle of the night in Zürich and she was unlikely to be awake for
another four or five hours. All of us had programs that connected us to
Sam by monitoring, and modifying, small portions of the defect:
altering a handful of precariously balanced truths of arithmetic,
wiggling the border between the two systems back and forth to encode
each transmitted bit. The three of us on the near side might have
communicated with each other in the same way, but on consideration we’d
decided that conventional cryptography was a safer way to conceal our
secret. The mere fact that communications data seemed to come from
nowhere had the potential to attract suspicion, so we’d gone so far as
to write software to send fake packets across the net to cover for our
otherwise inexplicable conversations with Sam; anyone but the most
diligent and resourceful of eavesdroppers would conclude that he was
addressing us from an internet café in Lithuania.
I was waiting for Yuen to reply, I scoured the logs where my knowledge
miner deposited results of marginal relevance, wondering if some flaw
in the criteria I’d given it might have left me with a blind spot. If
anyone, anywhere had announced their intention to carry out some kind
of calculation that might have led them to the defect, the news should
have been plastered across my desktop in flashing red letters within
seconds. Granted, most organizations with the necessary computing
resources were secretive by nature, but they were also unlikely to be
motivated to indulge in such a crazy stunt. Luminous itself had been
decommissioned in 2012; in principle, various national security
agencies, and even a few IT-centric businesses, now had enough silicon
to hunt down the defect if they’d really set their sights on it, but as
far as I knew Yuen, Alison, and I were still the only three people in
the world who were certain of its existence. The black budgets of even
the most profligate governments, the deep pockets of even the richest
tycoons, would not stretch far enough to take on the search as a long
shot, or an act of whimsy.
An IM window popped up with Alison’s face. She looked ragged. “What time is it there?” I asked.
“Early. Laura’s got colic.”
“Ah. Are you okay to talk?”
“Yeah, she’s asleep now.”
My email had been brief, so I filled her in on the details. She pondered the matter in silence for a while, yawning unashamedly.
only thing I can think of is some gossip I heard at a conference in
Rome a couple of months ago. It was a fourth-hand story about some guy
in New Zealand who thinks he’s found a way to test fundamental laws of
physics by doing computations in number theory.”
“Just random crackpot stuff, or . . . what?”
massaged her temples, as if trying to get more blood flowing to her
brain. “I don’t know, what I heard was too vague to make a judgment. I
gather he hasn’t tried to publish this anywhere, or even mentioned it
in blogs. I guess he just confided in a few people directly, one of
whom must have found it too amusing for them to keep their mouth shut.”
“Have you got a name?”
went off camera and rummaged for a while. “Tim Campbell,” she
announced. Her notes came through on the data channel. “He’s done
respectable work in combinatorics, algorithmic complexity,
optimization. I scoured the net, and there was no mention of this weird
stuff. I was meaning to email him, but I never got around to it.”
could understand why; that would have been about the time Laura was
born. I said, “I’m glad you still go to so many conferences in the
flesh. It’s easier in Europe, everything’s so close.”
“Ha! Don’t count on it continuing, Bruno. You might have to put your fat arse on a plane sometime yourself.”
“What about Yuen?”
frowned. “Didn’t I tell you? He’s been in hospital for a couple of
days. Pneumonia. I spoke to his daughter, he’s not in great shape.”
sorry.” Alison was much closer to him than I was; he’d been her
doctoral supervisor, so she’d known him long before the events that had
bound the three of us together.
Yuen was almost
eighty. That wasn’t yet ancient for a middle-class Chinese man who
could afford good medical care, but he would not be around forever.
said, “Are we crazy, trying to do this ourselves?” She knew what I
meant: liaising with Sam, managing the border, trying to keep the two
worlds talking but the two sides separate, safe and intact.
Alison replied, “Which government would you trust not to screw this up? Not to try to exploit it?”
But what’s the alternative? You pass the job on to Laura? Kate’s not
interested in having kids. So do I pick some young mathematician at
random to anoint as my successor?”
“Not at random, I’d hope.”
“You want me to advertise? ‘Must be proficient in number theory, familiar with Machiavelli, and own the complete boxed set of The West Wing?’ ”
shrugged. “When the time comes, find someone competent you can trust.
It’s a balance: the fewer people who know, the better, so long as there
are always enough of us that the knowledge doesn’t risk getting lost
“And this goes on generation after generation? Like some secret society? The Knights of the Arithmetic Inconsistency?”
“I’ll work on the crest.”
needed a better plan, but this wasn’t the time to argue about it. I
said, “I’ll contact this guy Campbell and let you know how it goes.”
“Okay. Good luck.” Her eyelids were starting to droop.
“Take care of yourself.”
managed an exhausted smile. “Are you saying that because you give a
damn, or because you don’t want to end up guarding the Grail all by
“Both, of course.”
“I have to fly to Wellington tomorrow.”
Kate put down the pasta-laden fork she’d raised halfway to her lips and gave me a puzzled frown. “That’s short notice.”
it’s a pain. It’s for the Bank of New Zealand. I have to do something
on-site with a secure machine, one they won’t let anyone access over
Her frown deepened. “When will you be back?”
not sure. It might not be until Monday. I can probably do most of the
work tomorrow, but there are certain things they restrict to the
weekends, when the branches are off-line. I don’t know if it will come
I hated lying to her, but I’d grown
accustomed to it. When we’d met, just a year after Shanghai, I could
still feel the scar on my arm where one of Industrial Algebra’s hired
thugs had tried to carve a data cache out of my body. At some point, as
our relationship deepened, I’d made up my mind that however close we
became, however much I trusted her, it would be safer for Kate if she
never knew anything about the defect.
“They can’t hire
someone local?” she suggested. I didn’t think she was suspicious, but
she was definitely annoyed. She worked long hours at the hospital, and
she only had every second weekend off; this would be one of them. We’d
made no specific plans, but it was part of our routine to spend this
I said, “I’m sure they could, but it’d
be hard to find someone at short notice. And I can’t tell them to shove
it, or I’ll lose the whole contract. It’s one weekend, it’s not the end
of the world.”
“No, it’s not the end of the world.” She finally lifted her fork again.
“Is the sauce okay?”
delicious, Bruno.” Her tone made it clear that no amount of culinary
effort would have been enough to compensate, so I might as well not
I watched her eat with a strange knot
growing in my stomach. Was this how spies felt, when they lied to their
families about their work? But my own secret sounded more like
something from a psychiatric ward. I was entrusted with the smooth
operation of a treaty that I, and two friends, had struck with an
invisible ghost world that coexisted with our own. The ghost world was
far from hostile, but the treaty was the most important in human
history, because either side had the power to annihilate the other so
thoroughly that it would make a nuclear holocaust seem like a pinprick.
University was in a hilltop suburb overlooking Wellington. I caught a
cable car, and arrived just in time for the Friday afternoon seminar.
Contriving an invitation to deliver a paper here myself would have been
difficult, but wangling permission to sit in as part of the audience
was easy; although I hadn’t been an academic for almost twenty years,
my ancient Ph.D and a trickle of publications, however tenuously
related to the topic of the seminar, were still enough to make me
I’d taken a gamble that Campbell would
attend—the topic was peripheral to his own research, official or
otherwise—so I was relieved to spot him in the audience, recognizing
him from a photo on the faculty web site. I’d emailed him straight
after I’d spoken to Alison, but his reply had been a polite brush-off:
he acknowledged that the work I’d heard about on the grapevine owed
something to the infamous search that Alison and I had launched, but he
wasn’t ready to make his own approach public.
through an hour on “Monoids and Control Theory,” trying to pay enough
attention that I wouldn’t make a fool of myself if the seminar
organizer quizzed me later on why I’d been sufficiently attracted to
the topic to interrupt my “sightseeing holiday” in order to attend.
When the seminar ended, the audience split into two streams: one
heading out of the building, the other moving into an adjoining room
where refreshments were on offer. I saw Campbell making for the open
air, and it was all I could do to contrive to get close enough to call
out to him without making a spectacle.
turned and scanned the room, probably expecting to see one of his
students wanting to beg for an extension on an assignment. I raised a
hand and approached him.
“Bruno Costanzo. I emailed you yesterday.”
course.” Campbell was a thin, pale man in his early thirties. He shook
my hand, but he was obviously taken aback. “You didn’t mention that you
were in Wellington.”
I made a dismissive gesture. “I
was going to, but then it seemed a bit presumptuous.” I didn’t spell it
out, I just left him to conclude that I was as ambivalent about this
whole inconsistency nonsense as he was.
If fate had brought us together, though, wouldn’t it be absurd not to make the most of it?
was going to grab some of those famous scones,” I said; the seminar
announcement on the web had made big promises for them. “Are you busy?”
“Umm. Just paperwork. I suppose I can put it off.”
we made our way into the tea room, I waffled on airily about my holiday
plans. I’d never actually been to New Zealand before, so I made it
clear that most of my itinerary still lay in the future. Campbell was
no more interested in the local geography and wildlife than I was; the
more I enthused, the more distant his gaze became. Once it was apparent
that he wasn’t going to cross-examine me on the finer points of various
hiking trails, I grabbed a buttered scone and switched subject abruptly.
thing is, I heard you’d devised a more efficient strategy for searching
for a defect.” I only just managed to stop myself from using the
definite article; it was a while since I’d spoken about it as if it
were still hypothetical. “You know the kind of computing power that Dr.
Tierney and I had to scrounge up?”
“Of course. I was just an undergraduate, but I heard about the search.”
you one of our volunteers?” I’d checked the records, and he wasn’t
listed, but people had had the option of registering anonymously.
The idea didn’t really grab me, at the time.” As he spoke, he seemed
more discomfited than the failure to donate his own resources twelve
years ago really warranted. I was beginning to suspect that he’d
actually been one of the people who’d found the whole tongue-in-cheek
conjecture that Alison and I had put forward to be unforgivably
foolish. We had never asked to be taken seriously—and we had even put
prominent links to all the worthy biomedical computing projects on our
web page, so that people knew there were far better ways to spend their
spare megaflops—but nonetheless, some mathematical/philosophical
stuffed shirts had spluttered with rage at the sheer impertinence and
naďvety of our hypothesis. Before things turned serious, it was the
entertainment value of that backlash that had made our efforts
“But now you’ve refined it somehow?” I
prompted him, doing my best to let him see that I felt no resentment at
the prospect of being outdone. In fact, the hypothesis itself had been
Alison’s, so even if there hadn’t been more important things than my
ego at stake, that really wasn’t a factor. As for the search algorithm,
I’d cobbled it together on a Sunday afternoon, as a joke, to call
Alison’s bluff. Instead, she’d called mine, and insisted that we
release it to the world.
Campbell glanced around to
see who was in earshot, but then perhaps it dawned on him that if the
news of his ideas had already reached Sydney via Rome and Zürich, the
battle to keep his reputation pristine in Wellington was probably lost.
said, “What you and Dr. Tierney suggested was that random processes in
the early universe might have included proofs of mutually contradictory
theorems about the integers, the idea being that no computation to
expose the inconsistency had yet had time to occur. Is that a fair
“One problem I have
with that is, I don’t see how it could lead to an inconsistency that
could be detected here and now. If the physical system A proved theorem
A, and the physical system B proved theorem B, then you might have
different regions of the universe obeying different axioms, but it’s
not as if there’s some universal mathematics textbook hovering around
outside spacetime, listing every theorem that’s ever been proved, which
our computers then consult in order to decide how to behave. The
behavior of a classical system is determined by its own particular
causal past. If we’re the descendants of a patch of the universe that
proved theorem A, our computers should be perfectly capable of disproving theorem B, whatever happened somewhere else fourteen billion years ago.”
nodded thoughtfully. “I can see what you’re getting at.” If you weren’t
going to accept full-blooded Platonism, in which there was a
kind of ghostly textbook listing the eternal truths of mathematics,
then a half-baked version where the book started out empty and was only
filled in line-by-line as various theorems were tested seemed like the
worst kind of compromise. In fact, when the far side had granted Yuen,
Alison, and I insight into their mathematics for a few minutes in
Shanghai, Yuen had proclaimed that the flow of mathematical information
did obey Einstein locality; there was no universal book of
truths, just records of the past sloshing around at lightspeed or less,
intermingling and competing.
I could hardly tell
Campbell, though, that not only did I know for a fact that a single
computer could prove both a theorem and its negation, but depending on
the order in which it attacked the calculations it could sometimes even
shift the boundary where one set of axioms failed and the other took
I said, “And yet you still believe it’s worth searching for an inconsistency?”
do,” he conceded. “Though I came to the idea from a very different
approach.” He hesitated, then picked up a scone from the table beside
“One rock, one apple, one scone. We have a clear
idea of what we mean by those phrases, though each one might encompass
ten-to-the-ten-to-the-thirty-something slightly different
configurations of matter. My ‘one scone’ is not the same as your ‘one
“You know how banks count large quantities of cash?”
weighing them?” In fact there were several other cross-checks as well,
but I could see where he was heading and I didn’t want to distract him
“Exactly. Suppose we tried to count
scones the same way: weigh the batch, divide by some nominal value,
then round to the nearest integer. The weight of any individual scone
varies so much that you could easily end up with a version of
arithmetic different from our own. If you ‘counted’ two separate
batches, then merged them and ‘counted’ them together, there’s no
guarantee that the result would agree with the ordinary process of
I said, “Clearly not. But digital computers don’t run on scones, and they don’t count bits by weighing them.”
“Bear with me,” Campbell replied. “It isn’t a perfect analogy, but I’m not as crazy as I sound. Suppose, now, that everything
we talk about as ‘one thing’ has a vast number of possible
configurations that we’re either ignoring deliberately, or are
literally incapable of distinguishing. Even something as simple as an
electron prepared in a certain quantum state.”
I said, “You’re talking about hidden variables now?”
“Of a kind, yes. Do you know about Gerard ’t Hooft’s models for deterministic quantum mechanics?”
“Only vaguely,” I admitted.
postulated fully deterministic degrees of freedom at the Planck scale,
with quantum states corresponding to equivalence classes containing
many different possible configurations. What’s more, all the ordinary
quantum states we prepare at an atomic level would be complex
superpositions of those primordial states, which allows him to get
around the Bell inequalities.” I frowned slightly; I more or less got
the picture, but I’d need to go away and read ’t Hooft’s papers.
Campbell said, “In a sense, the detailed physics isn’t all that important, so long as you accept that ‘one thing’ might not ever
be exactly the same as another ‘one thing,’ regardless of the kind of
objects we’re talking about. Given that supposition, physical processes
that seem to be rigorously equivalent to various arithmetic
operations can turn out not to be as reliable as you’d think. With
scone-weighing, the flaws are obvious, but I’m talking about the
potentially subtler results of misunderstanding the fundamental nature
“Hmm.” Though it was unlikely that anyone
else Campbell had confided in had taken these speculations as seriously
as I did, not only did I not want to seem a pushover, I honestly had no
idea whether anything he was saying bore the slightest connection to
I said, “It’s an interesting idea, but I still don’t see how it could speed up the hunt for inconsistencies.”
have a set of models,” he said, “which are constrained by the need to
agree with some of ’t Hooft’s ideas about the physics, and also by the
need to make arithmetic almost consistent for a very large
range of objects. From neutrinos to clusters of galaxies, basic
arithmetic involving the kinds of numbers we might encounter in
ordinary situations should work out in the usual way.” He laughed. “I
mean, that’s the world we’re living in, right?”
Some of us. “Yeah.”
the interesting thing is, I can’t make the physics work at all if the
arithmetic doesn’t run askew eventually—if there aren’t
trans-astronomical numbers where the physical representations no longer
capture the arithmetic perfectly. And each of my models lets me
predict, more or less, where those effects should begin to show up. By
starting with the fundamental physical laws, I can deduce a sequence of
calculations with large integers that ought to reveal an inconsistency,
when performed with pretty much any computer.”
you straight to the defect, with no need to search at all.” I’d let the
definite article slip out, but it hardly seemed to matter anymore.
the theory.” Campbell actually blushed slightly. “Well, when you say
‘no search,’ what’s involved really is a much smaller search. There are
still free parameters in my models; there are potentially billions of
possibilities to test.”
I grinned broadly, wondering if my expression looked as fake as it felt. “But no luck yet?”
“No.” He was beginning to become self-conscious again, glancing around to see who might be listening.
he lying to me? Keeping his results secret until he could verify them a
million more times, and then decide how best to explain them to
incredulous colleagues and an uncomprehending world? Or had whatever
he’d done that had lobbed a small grenade into Sam’s universe somehow
registered in Campbell’s own computer as arithmetic as usual, betraying
no evidence of the boundary he’d crossed? After all, the offending
cluster of propositions had obeyed our axioms, so perhaps
Campbell had managed to force them to do so without ever realizing that
they hadn’t in the past. His ideas were obviously close to the mark—and
I could no longer believe this was just a coincidence—but he seemed to
have no room in his theory for something that I knew for a fact:
arithmetic wasn’t merely inconsistent, it was dynamic. You could take its contradictions and slide them around like bumps in a carpet.
said, “Parts of the process aren’t easy to automate; there’s some
manual work to be done setting up the search for each broad class of
models. I’ve only been doing this in my spare time, so it could be a
while before I get around to examining all the possibilities.”
see.” If all of his calculations so far had produced just one hit on
the far side, it was conceivable that the rest would pass without
incident. He would publish a negative result ruling out an obscure
class of physical theories, and life would go on as normal on both
sides of the inconsistency.
What kind of weapons inspector would I be, though, to put my faith in that rosy supposition?
was looking fidgety, as if his administrative obligations were
beckoning. I said, “It’d be great to talk about this a bit more while
we’ve got the chance. Are you busy tonight? I’m staying at a
backpacker’s down in the city, but maybe you could recommend a
restaurant around here somewhere?”
He looked dubious
for a moment, but then an instinctive sense of hospitality seemed to
overcome his reservations. He said, “Let me check with my wife. We’re
not really into restaurants, but I was cooking tonight anyway, and
you’d be welcome to join us.”
house was a fifteen minute walk from the campus; at my request, we
detoured to a liquor store so I could buy a couple of bottles of wine
to accompany the meal. As I entered the house, my hand lingered on the
doorframe, depositing a small device that would assist me if I needed
to make an uninvited entry in the future.
wife, Bridget, was an organic chemist, who also taught at Victoria
University. The conversation over dinner was all about department
heads, budgets, and grant applications, and, despite having left
academia long ago, I had no trouble relating sympathetically to the
couple’s gripes. My hosts ensured that my wine glass never stayed empty
When we’d finished eating, Bridget excused
herself to make a call to her mother, who lived in a small town on the
south island. Campbell led me into his study and switched on a laptop
with fading keys that must have been twenty years old. Many households
had a computer like this: the machine that could no longer run the
latest trendy bloatware, but which still worked perfectly with its
Campbell turned his back to me as he
typed his password, and I was careful not to be seen even trying to
look. Then he opened some C++ files in an editor, and scrolled over
parts of his search algorithm.
I felt giddy, and it
wasn’t the wine; I’d filled my stomach with an over-the-counter
sobriety aid that turned ethanol into glucose and water faster than any
human being could imbibe it. I fervently hoped that Industrial Algebra
really had given up their pursuit; if I could get this close to
Campbell’s secrets in half a day, IA could be playing the stock market
with alternative arithmetic before the month was out, and peddling
inconsistency weapons to the Pentagon soon after.
did not have a photographic memory, and Campbell was just showing me
fragments anyway. I didn’t think he was deliberately taunting me; he
just wanted me to see that he had something concrete, that all his
claims about Planck scale physics and directed search strategies had
been more than hot air.
I said, “Wait! What’s that?”
He stopped hitting the PAGE DOWN key, and I pointed at a list of
variable declarations in the middle of the screen:
long int i1, i2, i3;
dark d1, d2, d3;
“long int” was a long integer, a quantity represented by twice as many
bits as usual. On this vintage machine, that was likely to be a total
of just sixty-four bits. “What the fuck is a ‘dark’?” I demanded. It
wasn’t how I’d normally speak to someone I’d only just met, but then, I
wasn’t meant to be sober.
Campbell laughed. “A dark integer. It’s a type I defined. It holds four thousand and ninety-six bits.”
“But why the name?”
matter, dark energy . . . dark integers. They’re all around us, but we
don’t usually see them, because they don’t quite play by the rules.”
Hairs rose on the back of my neck. I could not have described the infrastructure of Sam’s world more concisely myself.
shut down the laptop. I’d been looking for an opportunity to handle the
machine, however briefly, without arousing his suspicion, but that
clearly wasn’t going to happen, so as we walked out of the study I went
for plan B.
“I’m feeling kind of . . .” I sat down
abruptly on the floor of the hallway. After a moment, I fished my phone
out of my pocket and held it up to him. “Would you mind calling me a
“Yeah, sure.” He accepted the phone, and I
cradled my head in my arms. Before he could dial the number, I started
moaning softly. There was a long pause; he was probably weighing up the
embarrassment factor of various alternatives.
he said, “You can sleep here on the couch if you like.” I felt a
genuine pang of sympathy for him; if some clown I barely knew had
pulled a stunt like this on me, I would at least have made him promise
to foot the cleaning bills if he threw up in the middle of the night.
the middle of the night, I did make a trip to the bathroom, but I kept
the sound effects restrained. Halfway through, I walked quietly to the
study, crossed the room in the dark, and slapped a thin, transparent
patch over the adhesive label that a service company had placed on the
outside of the laptop years before. My addition would be invisible to
the naked eye, and it would take a scalpel to prise it off. The relay
that would communicate with the patch was larger, about the size of a
coat button; I stuck it behind a bookshelf. Unless Campbell was
planning to paint the room or put in new carpet, it would probably
remain undetected for a couple of years, and I’d already prepaid a two
year account with a local wireless internet provider.
woke not long after dawn, but this un-Bacchanalian early rising was no
risk to my cover; Campbell had left the curtains open so the full force
of the morning sun struck me in the face, a result that was almost
certainly deliberate. I tiptoed around the house for ten minutes or so,
not wanting to seem too organized if anyone was listening, then left a
scrawled note of thanks and apology on the coffee table by the couch,
before letting myself out and heading for the cable car stop.
in the city, I sat in a café opposite the backpacker’s hostel and
connected to the relay, which in turn had established a successful link
with the polymer circuitry of the laptop patch. When noon came and went
without Campbell logging on, I sent a message to Kate telling her that
I was stuck in the bank for at least another day.
passed the time browsing the news feeds and buying overpriced snacks;
half of the café’s other patrons were doing the same. Finally, just
after three o’clock, Campbell started up the laptop.
patch couldn’t read his disk drive, but it could pick up currents
flowing to and from the keyboard and the display, allowing it to deduce
everything he typed and everything he saw. Capturing his password was
easy. Better yet, once he was logged in he set about editing one of his
files, extending his search program to a new class of models. As he
scrolled back and forth, it wasn’t long before the patch’s screen shots
encompassed the entire contents of the file he was working on.
labored for more than two hours, debugging what he’d written, then set
the program running. This creaky old twentieth century machine, which
predated the whole internet-wide search for the defect, had already
scored one direct hit on the far side; I just hoped this new class of
models were all incompatible with the successful ones from a few days
Shortly afterward, the IR sensor in the patch
told me that Campbell had left the room. The patch could induce
currents in the keyboard connection; I could type into the machine as
if I was right there. I started a new process window. The laptop wasn’t
connected to the internet at all, except through my spyware, but it
took me only fifteen minutes to display and record everything there was
to see: a few library and header files that the main program depended
on, and the data logs listing all of the searches so far. It would not
have been hard to hack into the operating system and make provisions to
corrupt any future searches, but I decided to wait until I had a better
grasp of the whole situation. Even once I was back in Sydney, I’d be
able to eavesdrop whenever the laptop was in use, and intervene
whenever it was left unattended. I’d only stayed in Wellington in case
there’d been a need to return to Campbell’s house in person.
evening fell and I found myself with nothing urgent left to do, I
didn’t call Kate; it seemed wiser to let her assume that I was slaving
away in a windowless computer room. I left the café and lay on my bed
in the hostel. The dormitory was deserted; everyone else was out on the
I called Alison in Zürich and brought her up to
date. In the background, I could hear her husband, Philippe, trying to
comfort Laura in another room, calmly talking baby-talk in French while
his daughter wailed her head off.
intrigued. “Campbell’s theory can’t be perfect, but it must be close.
Maybe we’ll be able to find a way to make it fit in with the dynamics
we’ve seen.” In the ten years since we’d stumbled on the defect, all
our work on it had remained frustratingly empirical: running
calculations and observing their effects. We’d never come close to
finding any deep underlying principles.
“Do you think Sam knows all this?” she asked.
have no idea. If he did, I doubt he’d admit it.” Though it was Sam who
had given us a taste of far-side mathematics in Shanghai, that had
really just been a clip over the ear to let us know that what we were
trying to wipe out with Luminous was a civilization, not a wasteland.
After that near-disastrous first encounter, he had worked to establish
communications with us, learning our languages and happily listening to
the accounts we’d volunteered of our world, but he had not been equally
forthcoming in return. We knew next to nothing about far-side physics,
astronomy, biology, history, or culture. That there were living beings
occupying the same space as the Earth suggested that the two universes
were intimately coupled somehow, in spite of their mutual invisibility.
But Sam had hinted that life was much more common on his side of the
border than ours; when I’d told him that we seemed to be alone, at
least in the solar system, and were surrounded by light-years of
sterile vacuum, he’d taken to referring to our side as “Sparseland.”
said, “Either way, I think we should keep it to ourselves. The treaty
says we should do everything in our power to deal with any breach of
territory of which the other side informs us. We’re doing that. But
we’re not obliged to disclose the details of Campbell’s activities.”
true.” I wasn’t entirely happy with her suggestion, though. In spite of
the attitude Sam and his colleagues had taken—in which they assumed
that anything they told us might be exploited, might make them more
vulnerable—a part of me had always wondered if there was some gesture
of good faith we could make, some way to build trust. Since talking to
Campbell, in the back of my mind I’d been building up a faint hope that
his discovery might lead to an opportunity to prove, once and for all,
that our intentions were honorable.
Alison read my mood. She said, “Bruno, they’ve given us nothing.
Shanghai excuses a certain amount of caution, but we also know from
Shanghai that they could brush Luminous aside like a gnat. They have
enough computing power to crush us in an instant, and they still cling
to every strategic advantage they can get. Not to do the same ourselves
would just be stupid and irresponsible.”
“So you want
us to hold on to this secret weapon?” I was beginning to develop a
piercing headache. My usual way of dealing with the surreal
responsibility that had fallen on the three of us was to pretend that
it didn’t exist; having to think about it constantly for three days
straight meant more tension than I’d faced for a decade. “Is that what
it’s come down to? Our own version of the Cold War? Why don’t you just
march into NATO headquarters on Monday and hand over everything we
Alison said dryly, “Switzerland isn’t a member of NATO. The government here would probably charge me with treason.”
didn’t want to fight with her. “We should talk about this later. We
don’t even know exactly what we’ve got. I need to go through Campbell’s
files and confirm whether he really did what we think he did.”
“I’ll call you from Sydney.”
took me a while to make sense of everything I’d stolen from Campbell,
but eventually I was able to determine which calculations he’d
performed on each occasion recorded in his log files. Then I compared
the propositions that he’d tested with a rough, static map of the
defect; since the event Sam had reported had been deep within the far
side, there was no need to take account of the small fluctuations that
the border underwent over time.
If my analysis was
correct, late on Wednesday night Campbell’s calculations had landed in
the middle of far-side mathematics. He’d been telling me the truth,
though; he’d found nothing out of the ordinary there. Instead, the
thing he had been seeking had melted away before his gaze.
all the calculations Alison and I had done, only at the border had we
been able to force propositions to change their allegiance and obey our
axioms. It was as if Campbell had dived in from some higher dimension,
carrying a hosepipe that sprayed everything with the arithmetic we knew
For Sam and his colleagues, this was the
equivalent of a suitcase nuke appearing out of nowhere, as opposed to
the ICBMs they knew how to track and annihilate. Now Alison wanted us
to tell them, “Trust us, we’ve dealt with it,” without showing them the
weapon itself, without letting them see how it worked, without giving
them a chance to devise new defenses against it.
wanted us to have something up our sleeves, in case the hawks took over
the far side, and decided that Sparseland was a ghost world whose
lingering, baleful presence they could do without.
Saturday-night revelers began returning to the hostel, singing off-key
and puking enthusiastically. Maybe this was poetic justice for my own
faux-inebriation; if so, I was being repaid a thousandfold. I started
wishing I’d shelled out for classier accommodation, but since there was
no employer picking up my expenses, it was going to be hard enough
dealing with my lie to Kate without spending even more on the trip.
the arithmetic of scones; I knew how to make digital currency reproduce
like the marching brooms of the sorcerer’s apprentice. It might even
have been possible to milk the benefits without Sam noticing; I could
try to hide my far-sider trading behind the manipulations of the border
we used routinely to exchange messages.
I had no idea
how to contain the side-effects, though. I had no idea what else such
meddling would disrupt, how many people I might kill or maim in the
I buried my head beneath the pillows and
tried to find a way to get to sleep through the noise. I ended up
calculating powers of seven, a trick I hadn’t used since childhood. I’d
never been a prodigy at mental arithmetic, and the concentration
required to push on past the easy cases drained me far faster than any
physical labor. Two hundred and eighty-two million, four hundred and seventy-five thousand, two hundred and forty-nine.
The numbers rose into the stratosphere like bean stalks, until they
grew too high and tore themselves apart, leaving behind a cloud of
digits drifting through my skull like black confetti.
“The problem is under control,” I told Sam. “I’ve located the source, and I’ve taken steps to prevent a recurrence.”
you sure of that?” As he spoke, the three-holed torus on the screen
twisted restlessly. In fact I’d chosen the icon myself, and its
appearance wasn’t influenced by Sam at all, but it was impossible not
to project emotions onto its writhing.
I said, “I’m
certain that I know who was responsible for the incursion on Wednesday.
It was done without malice; in fact the person who did it doesn’t even
realize that he crossed the border. I’ve modified the operating system
on his computer so that it won’t allow him to do the same thing again;
if he tries, it will simply give him the same answers as before, but
this time the calculations won’t actually be performed.”
“That’s good to hear,” Sam said. “Can you describe these calculations?”
was as invisible to Sam as he was to me, but out of habit I tried to
keep my face composed. “I don’t see that as part of our agreement,” I
Sam was silent for a few seconds. “That’s
true, Bruno. But it might provide us with a greater sense of
reassurance if we knew what caused the breach in the first place.”
I said, “I understand. But we’ve made a decision.” We was Alison and I; Yuen was still in hospital, in no state to do anything. Alison and I, speaking for the world.
put your position to my colleagues,” he said. “We’re not your enemy,
Bruno.” His tone sounded regretful, and these nuances were under his control.
know that,” I replied. “Nor are we yours. Yet you’ve chosen to keep
most of the details of your world from us. We don’t view that as
evidence of hostility, so you have no grounds to complain if we keep a
few secrets of our own.”
“I’ll contact you again soon,” Sam said.
messenger window closed. I emailed an encrypted transcript to Alison,
then slumped across my desk. My head was throbbing, but the encounter
really hadn’t gone too badly. Of course Sam and his colleagues would
have preferred to know everything; of course they were going to be
disappointed and reproachful. That didn’t mean they were going to
abandon the benign policies of the last decade. The important thing was
that my assurance would prove to be reliable: the incursion would not
I had work to do, the kind that paid
bills. Somehow I summoned up the discipline to push the whole subject
aside and get on with a report on stochastic methods for resolving
distributed programming bottlenecks that I was supposed to be writing
for a company in Singapore.
Four hours later, when the
doorbell rang, I’d left my desk to raid the kitchen. I didn’t bother
checking the doorstep camera; I just walked down the hall and opened
Campbell said, “How are you, Bruno?”
“I’m fine. Why didn’t you tell me you were coming to Sydney?”
“Aren’t you going to ask me how I found your house?”
He held up his phone. There was a text message from me, or at least from my phone; it had SMS’d its GPS coordinates to him.
“Not bad,” I conceded.
believe they recently added ‘corrupting communications devices’ to the
list of terrorism-related offenses in Australia. You could probably get
me thrown into solitary confinement in a maximum security prison.”
“Only if you know at least ten words of Arabic.”
“Actually I spent a month in Egypt once, so anything’s possible. But I don’t think you really want to go to the police.”
I said, “Why don’t you come in?”
I showed him to the living room my mind was racing. Maybe he’d found
the relay behind the bookshelf, but surely not before I’d left his
house. Had he managed to get a virus into my phone remotely? I’d
thought my security was better than that.
Campbell said, “I’d like you to explain why you bugged my computer.”
“I’m growing increasingly unsure of that myself. The correct answer might be that you wanted me to.”
snorted. “That’s rich! I admit that I deliberately allowed a rumor to
start about my work, because I was curious as to why you and Alison
Tierney called off your search. I wanted to see if you’d come sniffing
around. As you did. But that was hardly an invitation to steal all my
“What was the point of the whole exercise for you, then, if not a way of stealing something from Alison and me?”
“You can hardly compare the two. I just wanted to confirm my suspicion that you actually found something.”
“And you believe that you’ve confirmed that?”
shook his head, but it was with amusement, not denial. I said, “Why are
you here? Do you think I’m going to publish your crackpot theory as my
own? I’m too old to get the Fields Medal, but maybe you think it’s
“Oh, I don’t think you’re interested in fame. As I said, I think you beat me to the prize a long time ago.”
rose to my feet abruptly; I could feel myself scowling, my fists
tightening. “So what’s the bottom line? You want to press charges
against me for the laptop? Go ahead. We can each get a fine in absentia.”
said, “I want to know exactly what was so important to you that you
crossed the Tasman, lied your way into my house, abused my hospitality,
and stole my files. I don’t think it was simply curiosity, or jealousy.
I think you found something ten years ago, and now you’re afraid my
work is going to put it at risk.”
I sat down again.
The rush of adrenaline I’d experienced at being cornered had
dissipated. I could almost hear Alison whispering in my ear, “Either
you kill him, Bruno, or you recruit him.” I had no intention of killing
anyone, but I wasn’t yet certain that these were the only two choices.
I said, “And if I tell you to mind your own business?”
shrugged. “Then I’ll work harder. I know you’ve screwed that laptop,
and maybe the other computers in my house, but I’m not so broke that I
can’t get a new machine.”
Which would be a hundred
times faster. He’d re-run every search, probably with wider parameter
ranges. The suitcase nuke from Sparseland that had started this whole
mess would detonate again, and for all I knew it could be ten times, a
hundred times, more powerful.
I said, “Have you ever wanted to join a secret society?”
Campbell gave an incredulous laugh. “No!”
“Neither did I. Too bad.”
told him everything. The discovery of the defect. Industrial Algebra’s
pursuit of the result. The epiphany in Shanghai. Sam establishing
contact. The treaty, the ten quiet years. Then the sudden jolt of his
own work, and the still-unfolding consequences.
was clearly shaken, but despite the fact that I’d confirmed his
original suspicion he wasn’t ready to take my word for the whole story.
knew better than to invite him into my office for a demonstration;
faking it there would have been trivial. We walked to the local
shopping center, and I handed him two hundred dollars to buy a new
notebook. I told him the kind of software he’d need to download,
without limiting his choice to any particular package. Then I gave him
some further instructions. Within half an hour, he had seen the defect
for himself, and nudged the border a short distance in each direction.
were sitting in the food hall, surrounded by boisterous teenagers who’d
just got out from school. Campbell was looking at me as if I’d seized a
toy machine gun from his hands, transformed it into solid metal, then
bashed him over the head with it.
I said, “Cheer up.
There was no war of the worlds after Shanghai; I think we’re going to
survive this, too.” After all these years, the chance to share the
burden with someone new was actually making me feel much more
“The defect is dynamic,” he muttered. “That changes everything.”
“You don’t say.”
Campbell scowled. “I don’t just mean the politics, the dangers. I’m talking about the underlying physical model.”
I hadn’t come close to examining that issue seriously; it had been
enough of a struggle coming to terms with his original calculations.
along, I’ve assumed that there were exact symmetries in the Planck
scale physics that accounted for a stable boundary between macroscopic
arithmetics. It was an artificial restriction, but I took it for
granted, because anything else seemed . . .”
He blinked and looked away, surveying the crowd of diners as if he had
no idea how he’d ended up among them. “I’m flying back in a few hours.”
“Does Bridget know why you came?”
I said, “No one else can know what I’ve told you. Not yet. The risks are too great, everything’s too fluid.”
“Yeah.” He met my gaze. He wasn’t just humoring me; he understood what people like IA might do.
the long term,” I said, “we’re going to have to find a way to make this
safe. To make everyone safe.” I’d never quite articulated that goal
before, but I was only just beginning to absorb the ramifications of
“How?” he wondered. “Do we want to build a wall, or do we want to tear one down?”
“I don’t know. The first thing we need is a better map, a better feel for the whole territory.”
hired a car at the airport in order to drive here and confront me; it
was parked in a side street close to my house. I walked him to it.
We shook hands before parting. I said, “Welcome to the reluctant cabal.”
Campbell winced. “Let’s find a way to change it from reluctant to redundant.”
the weeks that followed, Campbell worked on refinements to his theory,
emailing Alison and me every few days. Alison had taken my unilateral
decision to recruit Campbell with much more equanimity than I’d
expected. “Better to have him inside the tent,” was all she’d said.
proved to be an understatement. While the two of us soon caught up with
him on all the technicalities, it was clear that his intuition on the
subject, hard-won over many years of trial and error, was the key to
his spectacular progress now. Merely stealing his notes and his
algorithms would never have brought us so far.
the dynamic version of the theory took shape. As far as macroscopic
objects were concerned—and in this context, “macroscopic” stretched all
the way down to the quantum states of subatomic particles—all traces of
Platonic mathematics were banished. A “proof” concerning the integers
was just a class of physical processes, and the result of that proof
was neither read from, nor written to, any universal book of truths.
Rather, the agreement between proofs was simply a strong, but
imperfect, correlation between the different processes that counted as
proofs of the same thing. Those correlations arose from the way that
the primordial states of Planck-scale physics were carved
up—imperfectly—into subsystems that appeared to be distinct objects.
The truths of mathematics appeared
to be enduring and universal because they persisted with great
efficiency within the states of matter and space-time. But there was a
built-in flaw in the whole idealization of distinct objects, and the
point where the concept finally cracked open was the defect Alison and
I had found in our volunteers’ data, which appeared to any macroscopic
test as the border between contradictory mathematical systems.
derived a crude empirical rule which said that the border shifted when
a proposition’s neighbors outvoted it. If you managed to prove that
x+1=y+1 and x-1=y-1, then x=y became a sitting duck, even if it hadn’t
been true before. The consequences of Campbell’s search had shown that
the reality was more complex, and in his new model, the old border rule
became an approximation for a more subtle process, anchored in the
dynamics of primordial states that knew nothing of the arithmetic of
electrons and apples. The near-side arithmetic Campbell had blasted
into the far side hadn’t got there by besieging the target with
syllogisms; it had got there because he’d gone straight for a far
deeper failure in the whole idea of “integers” than Alison and I had
ever dreamed of.
Had Sam dreamed of it? I waited for
his next contact, but as the weeks passed he remained silent, and the
last thing I felt like doing was calling him myself. I had enough
people to lie to without adding him to the list.
asked me how work was going, and I waffled about the details of the
three uninspiring contracts I’d started recently. When I stopped
talking, she looked at me as if I’d just stammered my way through an
unconvincing denial of some unspoken crime. I wondered how my mixture
of concealed elation and fear was coming across to her. Was that how
the most passionate, conflicted adulterer would appear? I didn’t
actually reach the brink of confession, but I pictured myself
approaching it. I had less reason now to think that the secret would
bring her harm than when I’d first made my decision to keep her in the
dark. But then, what if I told her everything, and the next day
Campbell was kidnapped and tortured? If we were all being watched, and
the people doing it were good at their jobs, we’d only know about it
when it was too late.
Campbell’s emails dropped off
for a while, and I assumed he’d hit a roadblock. Sam had offered no
further complaints. Perhaps, I thought, this was the new status quo, the start of another quiet decade. I could live with that.
Then Campbell flung his second grenade. He reached me by IM and said, “I’ve started making maps.”
“Of the defect?” I replied.
“Of the planets.”
I stared at his image, uncomprehending.
“The far-side planets,” he said. “The physical worlds.”
bought himself some time on a geographically scattered set of processor
clusters. He was no longer repeating his dangerous incursions, of
course, but by playing around in the natural ebb and flow at the
border, he’d made some extraordinary discoveries.
and I had realized long ago that random “proofs” in the natural world
would influence what happened at the border, but Campbell’s theory made
that notion more precise. By looking at the exact timing of changes to
propositions at the border, measured in a dozen different computers
world-wide, he had set up a kind of . . . radar? CT machine? Whatever
you called it, it allowed him to deduce the locations where the
relevant natural processes were occurring, and his model allowed him to
distinguish between both near-side and far-side processes, and
processes in matter and those in vacuum. He could measure the density
of far-side matter out to a distance of several light-hours, and
crudely image nearby planets.
“Not just on the far
side,” he said. “I validated the technique by imaging our own planets.”
He sent me a data log, with comparisons to an online almanac. For
Jupiter, the farthest of the planets he’d located, the positions were
out by as much as a hundred thousand kilometers; not exactly GPS
quality, but that was a bit like complaining that your abacus couldn’t
tell north from north-west.
“Maybe that’s how Sam found us in Shanghai?” I wondered. “The same kind of thing, only more refined?”
Campbell said, “Possibly.”
“So what about the far-side planets?”
here’s the first interesting thing. None of the planets coincide with
ours. Nor does their sun with our sun.” He sent me an image of the
far-side system, one star and its six planets, overlaid on our own.
“But Sam’s time lags,” I protested, “when we communicate—”
“Make no sense if he’s too far away. Exactly. So he is not
living on any of these planets, and he’s not even in a natural orbit
around their star. He’s in powered flight, moving with the Earth. Which
suggests to me that they’ve known about us for much longer than
“Known about us,” I said, “but maybe they
still didn’t anticipate anything like Shanghai.” When we’d set Luminous
on to the task of eliminating the defect—not knowing that we were
threatening anyone—it had taken several minutes before the far side had
responded. Computers on board a spacecraft moving with the Earth would
have detected the assault quickly, but it might have taken the
recruitment of larger, planet-bound machines, minutes away at
lightspeed, to repel it.
Until I’d encountered
Campbell’s theories, my working assumption had been that Sam’s world
was like a hidden message encoded in the Earth, with the different
arithmetic giving different meanings to all the air, water, and rock
around us. But their matter was not bound to our matter; they didn’t
need our specks of dust or molecules of air to represent the dark
integers. The two worlds split apart at a much lower level; vacuum
could be rock, and rock, vacuum.
I said, “So do you want the Nobel for physics, or peace?”
Campbell smiled modestly. “Can I hold out for both?”
the answer I was looking for.” I couldn’t get the stupid Cold War
metaphors out of my brain: what would Sam’s hotheaded colleagues think,
if they knew that we were now flying spy planes over their territory?
Saying “screw them, they were doing it first!” might have been a fair
response, but it was not a particularly helpful one.
said, “We’re never going to match their Sputnik, unless you happen to
know a trustworthy billionaire who wants to help us launch a space
probe on a very strange trajectory. Everything we want to do has to
work from Earth.”
“I’ll tear up my letter to Richard Branson then, shall I?”
stared at the map of the far-side solar system. “There must be some
relative motion between their star and ours. It can’t have been this
close for all that long.”
“I don’t have enough
accuracy in my measurements to make a meaningful estimate of the
velocity,” Campbell said. “But I’ve done some crude estimates of the
distances between their stars, and it’s much smaller than ours. So it’s
not all that unlikely to find some star this close to us, even
if it’s unlikely to be the same one that was close a thousand years
ago. Then again, there might be a selection effect at work here: the
whole reason Sam’s civilization managed to notice us at all was because we weren’t shooting past them at a substantial fraction of lightspeed.”
“Okay. So maybe
this is their home system, but it could just as easily be an
expeditionary base for a team that’s been following our sun for
thousands of years.”
I said, “Where do we go with this?”
can’t increase the resolution much,” Campbell replied, “without buying
time on a lot more clusters.” It wasn’t that he needed much processing
power for the calculations, but there were minimum prices to be paid to
do anything at all, and what would give us clearer pictures would be
more computers, not more time on each one.
I said, “We
can’t risk asking for volunteers, like the old days. We’d have to lie
about what the download was for, and you can be certain that somebody
would reverse-engineer it and catch us out.”
slept on the problem, then woke with an idea at four a.m. and went to
my office, trying to flesh out the details before Campbell responded to
my email. He was bleary-eyed when the messenger window opened; it was
later in Wellington than in Sydney, but it looked as if he’d had as
little sleep as I had.
I said, “We use the internet.”
“I thought we decided that was too risky.”
“Not screensavers for volunteers; I’m talking about the internet itself.
We work out a way to do the calculations using nothing but data packets
and network routers. We bounce traffic all around the world, and we get
the geographical resolution for free.”
“You’ve got to be joking, Bruno—”
computing circuit can be built by stringing together enough NAND gates;
you think we can’t leverage packet switching into a NAND gate? But
that’s just the proof that it’s possible; I expect we can actually make
it a thousand times tighter.”
Campbell said, “I’m going to get some aspirin and come back.”
roped in Alison to help, but it still took us six weeks to get a
workable design, and another month to get it functioning. We ended up
exploiting authentication and error-correction protocols built into the
internet at several different layers; the heterogeneous approach not
only helped us do all the calculations we needed, but made our gentle
siphoning of computing power less likely to be detected and mistaken
for anything malicious. In fact we were “stealing” far less from the
routers and servers of the net than if we’d sat down for a hardcore 3D
multiplayer gaming session, but security systems had their own ideas
about what constituted fair use and what was suspicious. The most
important thing was not the size of the burden we imposed, but the
signature of our behavior.
Our new globe-spanning
arithmetical telescope generated pictures far sharper than before, with
kilometer-scale resolution out to a billion kilometers. This gave us
crude relief-maps of the far-side planets, revealing mountains on four
of them, and what might have been oceans on two of those four. If there
were any artificial structures, they were either too small to see, or
too subtle in their artificiality.
The relative motion
of our sun and the star these planets orbited turned out to be about
six kilometers per second. In the decade since Shanghai, the two solar
systems had changed their relative location by about two billion
kilometers. Wherever the computers were now that had fought with
Luminous to control the border, they certainly hadn’t been on any of
these planets at the time. Perhaps there were two ships, with one
following the Earth, and the other, heavier one saving fuel by merely
following the sun.
Yuen had finally recovered his health, and the full cabal held an IM-conference to discuss these results.
should be showing these to geologists, xenobiologists . . . everyone,”
Yuen lamented. He wasn’t making a serious proposal, but I shared his
sense of frustration.
Alison said, “What I regret most
is that we can’t rub Sam’s face in these pictures, just to show him
that we’re not as stupid as he thinks.”
“I imagine his own pictures are sharper,” Campbell replied.
is as you’d expect,” Alison retorted, “given a head start of a few
centuries. If they’re so brilliant on the far side, why do they need us to tell them what you did to jump the border?”
might have guessed precisely what I did,” he countered, “but they could
still be seeking confirmation. Perhaps what they really want is to rule
out the possibility that we’ve discovered something different,
something they’ve never even thought of.”
I gazed at
the false colors of one contoured sphere, imagining gray-blue oceans,
snow-topped mountains with alien forests, strange cities, wondrous
machines. Even if that was pure fantasy and this temporary neighbor was
barren, there had to be a living homeworld from which the ships that
pursued us had been launched.
After Shanghai, Sam and
his colleagues had chosen to keep us in the dark for ten years, but it
had been our own decision to cement the mistrust by holding on to the
secret of our accidental weapon. If they’d already guessed its nature,
then they might already have found a defense against it, in which case
our silence bought us no advantage at all to compensate for the
suspicion it engendered.
If that assumption was wrong, though?
Then handing over the details of Campbell’s work could be just what the
far-side hawks were waiting for, before raising their shields and
I said, “We need to make some plans. I
want to stay hopeful, I want to keep looking for the best way forward,
but we need to be prepared for the worst.”
that suggestion into something concrete required far more work than I’d
imagined; it was three months before the pieces started coming
together. When I finally shifted my gaze back to the everyday world, I
decided that I’d earned a break. Kate had a free weekend approaching; I
suggested a day in the Blue Mountains.
Her initial response was sarcastic, but when I persisted she softened a little, and finally agreed.
the drive out of the city, the chill that had developed between us
slowly began to thaw. We played JJJ on the car radio—laughing with
disbelief as we realized that today’s cutting-edge music consisted
mostly of cover versions and re-samplings of songs that had been hits
when we were in our twenties—and resurrected old running jokes from the
time when we’d first met.
As we wound our way into the
mountains, though, it proved impossible simply to turn back the clock.
Kate said, “Whoever you’ve been working for these last few months, can
you put them on your blacklist?”
I laughed. “That will
scare them.” I switched to my best Brando voice. “You’re on Bruno
Costanzo’s blacklist. You’ll never run distributed software efficiently
in this town again.”
She said, “I’m serious. I don’t know what’s so stressful about the work, or the people, but it’s really screwing you up.”
could have made her a promise, but it would have been hard enough to
sound sincere as I spoke the words, let alone live up to them. I said,
“Beggars can’t be choosers.”
She shook her head, her
mouth tensed in frustration. “If you really want a heart attack, fine.
But don’t pretend that it’s all about money. We’re never that broke,
and we’re never that rich. Unless it’s all going into your account in
It took me a few seconds to convince myself
that this was nothing more than a throwaway reference to Swiss banks.
Kate knew about Alison, knew that we’d once been close, knew that we
still kept in touch. She had plenty of male friends from her own past,
and they all lived in Sydney; for more than five years, Alison and I
hadn’t even set foot on the same continent.
the car, then walked along a scenic trail for an hour, mostly in
silence. We found a spot by a stream, with tiered rocks smoothed by
some ancient river, and ate the lunch I’d packed.
out into the blue haze of the densely wooded valley below, I couldn’t
keep the image of the crowded skies of the far side from my mind. A
dazzling richness surrounded us: alien worlds, alien life, alien
culture. There had to be a way to end our mutual suspicion, and work
toward a genuine exchange of knowledge.
As we started
back toward the car, I turned to Kate. “I know I’ve neglected you,” I
said. “I’ve been through a rough patch, but everything’s going to
change. I’m going to make things right.”
I was prepared for a withering rebuff, but for a long time she was silent. Then she nodded slightly and said, “Okay.”
she reached across and took my hand, my wrist began vibrating. I’d
buckled to the pressure and bought a watch that shackled me to the net
twenty-four hours a day.
I freed my hand from Kate’s
and lifted the watch to my face. The bandwidth reaching me out in the
sticks wasn’t enough for video, but a stored snapshot of Alison
appeared on the screen.
“This is for emergencies only,” I snarled.
out a news feed,” she replied. The acoustics were focused on my ears;
Kate would get nothing but the bad-hearing-aid-at-a-party impression
that made so many people want to punch their fellow commuters on trains.
“Why don’t you just summarize whatever it is I’m meant to have noticed?”
computing systems were going haywire, to an extent that was already
being described as terrorism. Most trading was closed for the weekend,
but some experts were predicting the crash of the century, come Monday.
wondered if the cabal itself was to blame; if we’d inadvertently
corrupted the whole internet by coupling its behavior to the defect.
That was nonsense, though. Half the transactions being garbled were
taking place on secure, interbank networks that shared no hardware with
our global computer. This was coming from the far side.
“Have you contacted Sam?” I asked her.
“I can’t raise him.”
are you going?” Kate shouted angrily. I’d unconsciously broken into a
jog; I wanted to get back to the car, back to the city, back to my
I stopped and turned to her. “Run with me? Please? This is important.”
“You’re joking! I’ve spent half a day hiking, I’m not running anywhere!”
hesitated, fantasizing for a moment that I could sit beneath a gum tree
and orchestrate everything with my Dick Tracy watch before its battery
I said, “You’d better call a taxi when you get to the road.”
“You’re taking the car?” Kate stared at me, incredulous. “You piece of shit!”
“I’m sorry.” I tossed my backpack on the ground and started sprinting.
“We need to deploy,” I told Alison.
“I know,” she said. “We’ve already started.”
was the right decision, but hearing it still loosened my bowels far
more than the realization that the far side were attacking us. Whatever
their motives, at least they were unlikely to do more harm than they
intended. I was much less confident about our own abilities.
“Keep trying to reach Sam,” I insisted. “This is a thousand times more useful if they know about it.”
Alison said, “I guess this isn’t the time for Dr. Strangelove jokes.”
the last three months, we’d worked out a way to augment our internet
“telescope” software to launch a barrage of Campbell-style attacks on
far-side propositions if it saw our own mathematics being encroached
upon. The software couldn’t protect the whole border, but there were
millions of individual trigger points, forming a randomly shifting
minefield. The plan had been to buy ourselves some security, without
ever reaching the point of actual retaliation. We’d been waiting to
complete a final round of tests before unleashing this version live on
the net, but it would only take a matter of minutes to get it up and
“Anything being hit besides financials?” I asked.
“Not that I’m picking up.”
the far side was deliberately targeting the markets, that was
infinitely preferable to the alternative: that financial systems had
simply been the most fragile objects in the path of a much broader
assault. Most modern engineering and aeronautical systems were more
interested in resorting to fall-backs than agonizing over their
failures. A bank’s computer might declare itself irretrievably
compromised and shut down completely, the instant certain totals failed
to reconcile; those in a chemical plant or an airliner would be
designed to fail more gracefully, trying simpler alternatives and
bringing all available humans into the loop.
I said, “Yuen and Tim—?”
“Both on board,” Alison confirmed. “Monitoring the deployment, ready to tweak the software if necessary.”
“Good. You really won’t need me at all, then, will you?”
reply dissolved into digital noise, and the connection cut out. I
refused to read anything sinister into that; given my location, I was
lucky to have any coverage at all. I ran faster, trying not to think
about the time in Shanghai when Sam had taken a mathematical scalpel to
all of our brains. Luminous had been screaming out our position like a
beacon; we would not be so easy to locate this time. Still, with a
cruder approach, the hawks could take a hatchet to everyone’s head. Would they go that far?
Only if this was meant as much more than a threat, much more than
intimidation to make us hand over Campbell’s algorithm. Only if this
was the end game: no warning, no negotiations, just Sparseland wiped
off the map forever.
Fifteen minutes after Alison’s
call, I reached the car. Apart from the entertainment console it didn’t
contain a single microchip; I remembered the salesman laughing when I’d
queried that twice. “What are you afraid of? Y3K?” The engine started
I had an ancient secondhand laptop in the
trunk; I put it beside me on the passenger seat and started booting it
up while I drove out on to the access road, heading for the highway.
Alison and I had worked for a fortnight on a stripped-down operating
system, as simple and robust as possible, to run on these old
computers; if the far side kept reaching down from the arithmetic
stratosphere, these would be like concrete bunkers compared to the
glass skyscrapers of more modern machines. The four of us would also be
running different versions of the OS, on CPUs with different
instruction sets; our bunkers were scattered mathematically as well as
As I drove on to the highway, my watch stuttered back to life. Alison said, “Bruno? Can you hear me?”
“Three passenger jets have crashed,” she said. “Poland, Indonesia, South Africa.”
was dazed. Ten years before, when I’d tried to bulldoze his whole
mathematical world into the sea, Sam had spared my life. Now the far
side was slaughtering innocents.
“Is our minefield up?”
“It’s been up for ten minutes, but nothing’s tripped it yet.”
“You think they’re steering through it?”
hesitated. “I don’t see how. There’s no way to predict a safe path.” We
were using a quantum noise server to randomize the propositions we
I said, “We should trigger it manually. One
counter-strike to start with, to give them something to think about.” I
was still hoping that the downed jets were unintended, but we had no
choice but to retaliate.
“Yeah.” Alison’s image was
live now; I saw her reach down for her mouse. She said, “It’s not
responding. The net’s too degraded.” All the fancy algorithms that the
routers used, and that we’d leveraged so successfully for our imaging
software, were turning them into paperweights. The internet was robust
against high levels of transmission noise and the loss of thousands of
connections, but not against the decay of arithmetic itself.
watch went dead. I looked to the laptop; it was still working. I
reached over and hit a single hotkey, launching a program that would
try to reach Alison and the others the same way we’d talked to Sam: by
modulating part of the border. In theory, the hawks might have moved
the whole border—in which case we were screwed—but the border was vast,
and it made more sense for them to target their computing resources on
the specific needs of the assault itself.
A small icon appeared on the laptop’s screen, a single letter A in reversed monochrome. I said, “Is this working?”
Alison replied. The icon blinked out, then came back again. We were
doing a Hedy Lamarr, hopping rapidly over a predetermined sequence of
border points to minimize the chance of detection. Some of those points
would be missing, but it looked as if enough of them remained intact.
A was joined by a Y and a T. The whole cabal was online now, whatever
that was worth. What we needed was S, but S was not answering.
said grimly, “I heard about the planes. I’ve started an attack.” The
tactic we had agreed upon was to take turns running different variants
of Campbell’s border-jumping algorithm from our scattered machines.
said, “The miracle is that they’re not hitting us the same way we’re
hitting them. They’re just pushing down part of the border with the old
voting method, step by step. If we’d given them what they’d asked for,
we’d all be dead by now.”
“Maybe not,” Yuen replied.
“I’m only halfway through a proof, but I’m 90 percent sure that Tim’s
method is asymmetrical. It only works in one direction. Even if we’d
told them about it, they couldn’t have turned it against us.”
opened my mouth to argue, but if Yuen was right that made perfect
sense. The far side had probably been working on the same branch of
mathematics for centuries; if there had been an equivalent weapon that
could be used from their vantage point, they would have discovered it
My machine had synchronized with Campbell’s,
and it took over the assault automatically. We had no real idea what we
were hitting, except that the propositions were further from the
border, describing far simpler arithmetic on the dark integers than
anything of ours that the far side had yet touched. Were we crippling machines? Taking lives? I was torn between a triumphant vision of retribution, and a sense of shame that we’d allowed it to come to this.
hundred meters or so, I passed another car sitting motionless by the
side of the highway. I was far from the only person still driving, but
I had a feeling Kate wouldn’t have much luck getting a taxi. She had
water in her backpack, and there was a small shelter at the spot where
we’d parked. There was little to be gained by reaching my office now;
the laptop could do everything that mattered, and I could run it from
the car battery if necessary. If I turned around and went back for
Kate, though, I’d have so much explaining to do that there’d be no time
for anything else.
I switched on the car radio, but
either its digital signal processor was too sophisticated for its own
good, or all the local stations were out.
“Anyone still getting news?” I asked.
still have radio,” Campbell replied. “No TV, no internet. Landlines and
mobiles here are dead.” It was the same for Alison and Yuen. There’d
been no more reports of disasters on the radio, but the stations were
probably as isolated now as their listeners. Ham operators would still
be calling each other, but journalists and newsrooms would not be in
the loop. I didn’t want to think about the contingency plans that might
have been in place, given ten years’ preparation and an informed
By the time I reached Penrith there were
so many abandoned cars that the remaining traffic was almost
gridlocked. I decided not to even try to reach home. I didn’t know if
Sam had literally scanned my brain in Shanghai and used that to target
what he’d done to me then, and whether or not he could use the same
neuroanatomical information against me now, wherever I was, but staying
away from my usual haunts seemed like one more small advantage to cling
I found a gas station, and it was giving priority
to customers with functioning cars over hoarders who’d appeared on foot
with empty cans. Their EFTPOS wasn’t working, but I had enough cash for
the gas and some chocolate bars.
As dusk fell the
streetlights came on; the traffic lights had never stopped working. All
four laptops were holding up, hurling their grenades into the far side.
The closer the attack front came to simple arithmetic, the more
resistance it would face from natural processes voting at the border
for near-side results. Our enemy had their supercomputers; we had every
atom of the Earth, following its billion-year-old version of the truth.
had modeled this scenario. The sheer arithmetical inertia of all that
matter would buy us time, but in the long run a coherent, sustained,
computational attack could still force its way through.
would we die? Losing consciousness first, feeling no pain? Or was the
brain more robust than that? Would all the cells of our bodies start
committing apoptosis, once their biochemical errors mounted up beyond
repair? Maybe it would be just like radiation sickness. We’d be burned
by decaying arithmetic, just as if it was nuclear fire.
laptop beeped. I swerved off the road and parked on a stretch of
concrete beside a dark shopfront. A new icon had appeared on the
screen: the letter S.
Sam said, “Bruno, this was not my decision.”
“I believe you,” I said. “But if you’re just a messenger now, what’s your message?”
“If you give us what we asked for, we’ll stop the attack.”
“We’re hurting you, aren’t we?”
“We know we’re hurting you,” Sam replied. Point taken: we were guessing, firing blind. He didn’t have to ask about the damage we’d suffered.
steeled myself, and followed the script the cabal had agreed upon.
“We’ll give you the algorithm, but only if you retreat back to the old
border, and then seal it.”
Sam was silent for four long heartbeats.
think you know what I mean.” In Shanghai, when we’d used Luminous to
try to ensure that Industrial Algebra could not exploit the defect,
we’d contemplated trying to seal the border rather than eliminating the
defect altogether. The voting effect could only shift the border if it
was crinkled in such a way that propositions on one side could be
outnumbered by those on the other side. It was possible—given enough
time and computing power—to smooth the border, to iron it flat. Once
that was done, everywhere, the whole thing would become immovable. No
force in the universe could shift it again.
Sam said, “You want to leave us with no weapon against you, while you still have the power to harm us.”
“We won’t have that power for long. Once you know exactly what we’re using, you’ll find a way to block it.”
There was a long pause. Then, “Stop your attacks on us, and we’ll consider your proposal.”
“We’ll stop our attacks when you pull the border back to the point where our lives are no longer at risk.”
would you even know that we’ve done that?” Sam replied. I wasn’t sure
if the condescension was in his tone or just his words, but either way
I welcomed it. The lower the far side’s opinion of our abilities, the
more attractive the deal became for them.
“Then you’d better back up far enough for all our communications
systems to recover. When I can get news reports and see that there are
no more planes going down, no power plants exploding, then we’ll start
Silence again, stretching out beyond
mere hesitancy. His icon was still there, though, the S unblinking. I
clutched at my shoulder, hoping that the burning pain was just tension
in the muscle.
Finally: “All right. We agree. We’ll start shifting the border.”
drove around looking for an all-night convenience store that might have
had an old analog TV sitting in a corner to keep the cashier awake—that
seemed like a good bet to start working long before the wireless
connection to my laptop—but Campbell beat me to it. New Zealand radio
and TV were reporting that the “digital blackout” appeared to be
lifting, and ten minutes later Alison announced that she had internet
access. A lot of the major servers were still down, or their sites
weirdly garbled, but Reuters was starting to post updates on the crisis.
had kept his word, so we halted the counter-strikes. Alison read from
the Reuters site as the news came in. Seventeen planes had crashed, and
four trains. There’d been fatalities at an oil refinery, and half a
dozen manufacturing plants. One analyst put the global death toll at
five thousand and rising.
I muted the microphone on my
laptop and spent thirty seconds shouting obscenities and punching the
dashboard. Then I rejoined the cabal.
Yuen said, “I’ve
been reviewing my notes. If my instinct is worth anything, the theorem
I mentioned before is correct: if the border is sealed, they’ll have no
way to touch us.”
“What about the upside for them?”
Alison asked. “Do you think they can protect themselves against Tim’s
algorithm, once they understand it?”
“Yes and no. Any cluster of near-side truth values it injects into the
far side will have a non-smooth border, so they’ll be able to remove it
with sheer computing power. In that sense, they’ll never be
defenseless. But I don’t see how there’s anything they can do to
prevent the attacks in the first place.”
“Short of wiping us out,” Campbell said.
I heard an infant sobbing. Alison said, “That’s Laura. I’m alone here. Give me five minutes.”
buried my head in my arms. I still had no idea what the right course
would have been. If we’d handed over Campbell’s algorithm immediately,
might the good will that bought us have averted the war? Or would the
same attack merely have come sooner? What criminal vanity had ever made
the three of us think we could shoulder this responsibility on our own?
Five thousand people were dead. The hawks who had taken over on the far
side would weigh up our offer, and decide that they had no choice but
to fight on.
And if the reluctant cabal had passed its
burden to Canberra, to Zürich, to Beijing? Would there really have been
peace? Or was I just wishing that there had been more hands steeped in
the same blood, to share the guilt around?
The idea came from nowhere, sweeping away every other thought. I said, “Is there any reason why the far side has to stay connected?”
“Connected to what?” Campbell asked.
to itself. Connected topologically. They should be able to send down a
spike, then withdraw it, but leave behind a bubble of altered truth
values: a kind of outpost, sitting within the near side, with a
perfect, smooth border making it impregnable. Right?”
Yuen said, “Perhaps. With both sides collaborating on the construction, that might be possible.”
the question is, can we find a place where we can do that so that it
kills off the chance to use Tim’s method completely—without crippling
any process that we need just to survive?”
Bruno!” Campbell exclaimed happily. “We give them one small Achilles
tendon to slice . . . and then they’ve got nothing to fear from us!”
Yuen said, “A watertight proof of something like that is going to take weeks, months.”
we’d better start work. And we’d better feed Sam the first plausible
conjecture we get, so they can use their own resources to help us with
Alison came back online and greeted the
suggestion with cautious approval. I drove around until I found a quiet
coffee shop. Electronic banking still wasn’t working, and I had no cash
left, but the waiter agreed to take my credit card number and a signed
authority for a deduction of one hundred dollars; whatever I didn’t eat
and drink would be his tip.
I sat in the café,
blanking out the world, steeping myself in the mathematics. Sometimes
the four of us worked on separate tasks; sometimes we paired up,
dragging each other out of dead ends and ruts. There were an infinite
number of variations that could be made to Campbell’s algorithm, but
hour by hour we whittled away at the concept, finding the common ground
that no version of the weapon could do without.
By four in the morning, we had a strong conjecture. I called Sam, and explained what we were hoping to achieve.
He said, “This is a good idea. We’ll consider it.”
café closed. I sat in the car for a while, drained and numb, then I
called Kate to find out where she was. A couple had given her a lift
almost as far as Penrith, and when their car failed she’d walked the
rest of the way home.
close to four days, I spent most of my waking hours just sitting at my
desk, watching as a wave of red inched its way across a map of the
defect. The change of hue was not being rendered lightly; before each
pixel turned red, twelve separate computers needed to confirm that the
region of the border it represented was flat.
fifth day, Sam shut off his computers and allowed us to mount an attack
from our side on the narrow corridor linking the bulk of the far side
with the small enclave that now surrounded our Achilles’ Heel. We
wouldn’t have suffered any real loss of essential arithmetic if this
slender thread had remained, but keeping the corridor both small and
impregnable had turned out to be impossible. The original plan was the
only route to finality: to seal the border perfectly, the far side
proper could not remain linked to its offshoot.
next stage, the two sides worked together to seal the enclave
completely, polishing the scar where its umbilical had been sheared
away. When that task was complete, the map showed it as a single
burnished ruby. No known process could reshape it now. Campbell’s
method could have breached its border without touching it, reaching
inside to reclaim it from within—but Campbell’s method was exactly what
this jewel ruled out.
At the other end of the vanished
umbilical, Sam’s machines set to work smoothing away the blemish. By
early evening that, too, was done.
Only one tiny flaw
in the border remained now: the handful of propositions that enabled
communication between the two sides. The cabal had debated the fate of
this for hours. So long as this small wrinkle persisted, in principle
it could be used to unravel everything, to mobilize the entire border
again. It was true that, compared to the border as a whole, it would be
relatively easy to monitor and defend such a small site, but a
sustained burst of brute-force computing from either side could still
overpower any resistance and exploit it.
In the end,
Sam’s political masters had made the decision for us. What they had
always aspired to was certainty, and even if their strength favored
them, this wasn’t a gamble they were prepared to take.
I said, “Good luck with the future.”
luck to Sparseland,” Sam replied. I believed he’d tried to hold out
against the hawks, but I’d never been certain of his friendship. When
his icon faded from my screen, I felt more relief than regret.
learned the hard way not to assume that anything was permanent. Perhaps
in a thousand years, someone would discover that Campbell’s model was
just an approximation to something deeper, and find a way to fracture
these allegedly perfect walls. With any luck, by then both sides might
also be better prepared to find a way to co-exist.
found Kate sitting in the kitchen. I said, “I can answer your questions
now, if that’s what you want.” On the morning after the disaster, I’d
promised her this time would come—within weeks, not months—and she’d
agreed to stay with me until it did.
She thought for a while.
“Did you have something to do with what happened last week?”
you saying you unleashed the virus? You’re the terrorist they’re
looking for?” To my great relief, she asked this in roughly the tone
she might have used if I’d claimed to be Genghis Khan.
I’m not the cause of what happened. It was my job to try and stop it,
and I failed. But it wasn’t any kind of computer virus.”
She searched my face. “What was it, then? Can you explain that to me?”
“It’s a long story.”
“I don’t care. We’ve got all night.”
I said, “It started in university. With an idea of Alison’s. One
brilliant, beautiful, crazy idea.”
looked away, her face flushing, as if I’d said something deliberately
humiliating. She knew I was not a mass murderer. But there were other
things about me of which she was less sure.
“The story starts with Alison,” I said. “But it ends here, with you.”