Authors: Greg Egan
Tags: #Science Fiction
"Fair enough," I conceded warily. "But that doesn't mean
"And maybe Andrew Wiles's brain
and body, and notepaper
comprised the first physical sy
tem whose behavior depended on the theorem being true or false. But I don't think human actions have any special role
. . .
and if some swarm of quarks had done the same thing blindly, fifteen billion years before
executed some purely random interaction that just happened to test the conjecture in some way
would have constructed FLT long before Wiles. We'll never know."
I opened my mouth to complain that no swarm of quarks could have tested the infinite number of cases encompassed by the theorem
but I caught myself just in time. That was true
but it ha
n't stopped Wiles. A finite sequence of logical steps linked the axioms of number theory
which included some simple generalities about
to Fermat's own sweeping assertion. And if a mathematician could test those logical steps by manipulating a finite number of physical o
jects for a finite amount of time
whether they were pencil marks on paper, or neurotransmitters in his or her brain
then all kinds of physical systems could, in theory, mimic the structure of the proof
. . .
with or without any awareness of what it was they were "proving."
I leant back on the bench and mimed tearing out hair. "If I wasn't a die-hard Platonist before, you're forcing me into it! Fermat's Last Theorem didn't
to be proved by anyone
bled on by any random swarm of quarks. If it's true, it was always true. Everything implied by a given set of axioms is logically connected to them, timelessly, eternally
. . .
even if the links couldn't be traced by people
in the lifetime of the universe."
Alison was having none of this; every mention of
timeless and eternal truths
brought a faint smile to the comer of her mouth, as if I was affirming my belief in Santa Claus. She said, "So who, or what, pushed the consequences of 'There exists an entity called zero' and 'Every X has a su
all the way to Fermat's Last Theorem and beyond, before the universe had a chance to test out any of it?"
I stood my ground. "What's joined by logic is just
. . .
Nothing has to happen
quences don't have to be 'pushed' into existence by anyone, or anything. Or do you imagine that the first events after the Big Bang, the first wild jitters of the quark-gluon plasma, stopped to fill in all the logical gaps? You think the quarks reasoned: well, so far we've done A and
but now we mustn't
do D, because D would be logically inconsistent with the other mathematics we've 'invented' so far
even if it would take a five-hundred-thousand-page proof to spell out the inconsistency?"
Alison thought it over. "No. But what if event D took place, regardless? What if the mathematics it implied
logically inconsistent with the rest
but it went ahead and happened anyway
. . .
cause the universe was too young to have computed the fact that there was any discrepancy?"
I must have sat and stared at her, open-mouthed, for about ten seconds. Given the orthodoxies we'd spent the last two-and-a-half years absorbing, this was a seriously outrageous statement.
"You're claiming that
. . .
might be strewn with primordial defects in consistency? Like space might be strewn with cosmic strings?"
"Exactly." She stared back at me, feigning nonchalance. "If space-time doesn't join up with itself smoothly, everywhere
. . .
why should mathematical logic?"
I almost choked.
'Where do I begin? What happens
when some physical system tries to link theorems across the defect? If theorem D has been rendered 'true' by some over-eager quarks, what happens when we program a computer to disprove it? When the software goes through all the logical steps that link A, B, and
which the quarks have also made true
to the contradiction, the dreaded not-D
. . .
does it succeed, or doesn't it?"
Alison side-stepped the question. "Suppose they're both true: D and not-D. Sounds like the end of mathematics, doesn't it? The whole system falls apart, instantly. From D and not-D together you can prove anything you like: one equals zero, day equals night. But that's just the boring-old-fart Platonist view
where logic travels faster than light, and computation takes no time at all. People live with omega-inconsistent theories, don't they?"
Omega-inconsistent number theories were nonstandard versions of arithmetic, based on axioms that "almost" contradicted each other
their saving grace being that the contradictions could only show up in "infinitely long proofs" (which were formally disallowed, quite apart from being phys
cally impossible). That was perfectly respectable modem mathematics
but Alison seemed pr
pared to replace "infinitely long" with just plain "long"
as if the difference hardly mattered, in practice.
I said, "Let me get this straight. What you're talking about is taking ordinary arithmetic
no weird counter-intuitive axioms, just the stuff every ten-year-old
and proving that it's i
consistent, in a finite number of steps?"
She nodded blithely. "Finite, but large. So the contradiction would rarely have any physical man
it would be 'computationally distant' from everyday calculations, and everyday physical events. I mean
. . .
one cosmic string, somewhere out there, doesn't destroy the universe, does it? It does no harm to anyone.
I laughed drily. "So long as you don't get too close. So long as you don't tow it back to the solar system and let it twitch around slicing up planets." "Exactly."
I glanced at my watch. "Time to come down to Earth, I think. You know we're meeting Julia and Ramesh
Alison sighed theatrically. "I know, I know. And this would bore them witless, poor things
so the subject's closed, I promise.
She added wickedly, "Humanities students are so
We set off across the tranquil leafy campus. Alison kept her word, and we walked in silence; ca
rying on the argument up to the last minute would have made it even harder to avoid the topic once we were in polite company.
Halfway to the cafeteria, though, I couldn't help myself.
"If someone ever
program a computer to follow a chain of inferences across the defect
. . .
what do you claim would actually happen? When the end result of all those simple, trustworthy logical steps finally popped up on the screen
which group of primordial quarks would win the battle? And please don't tell me that the whole computer just conveniently vanishes."
Alison smiled, tongue-in-cheek at last. "Get real, Bruno. How can you expect me to answer that, when the mathematics needed to predict the result doesn't even
yet? Nothing I could say would be true or false
until someone's gone ahead and done the experiment."
I spent most of the day trying to convince myself that I wasn't being followed by some accomplice (or rival) of the surgeon, who might have been lurking outside the hotel. There was something disturbingly Kafkaesque about trying to lose a tail who might or might not have been real: no pa
ticular face I could search for in the crowd, just the abstract idea of a pursuer. It was too late to think about plastic surgery to make me look Han Chinese
Alison had raised this as a serious suggestion, back in Vietnam
but Shanghai had over a million foreign residents, so with care even an Anglophone of Italian descent should have been able to vanish.
Whether or not I was up to the task was another matter.
I tried joining the ant-trails of the tourists, following the path of least resistance from the insane crush of the Yuyuan Bazaar (where racks bursting with tencent watch-PCs, mood-sensitive contact lenses, and the latest karaoke vocal implants, sat beside bamboo cages of live ducks and pigeons) to the one-time residence of Sun Yatsen (whose personality cult was currently unde
going a mini-series-led revival on Star TV, advertised on ten thousand buses and ten times as many T-shirts). From the tomb of the writer Lu Xun ("Always think and study
. . .
visit the general then visit the victims, see the realities of your time with open eyes"
no prime time for
to the Hongkou McDonald's (where they were giving away small plastic Andy Warhol figurines, for re
sons I couldn't fathom). I mimed leisurely window-shopping between the shrines, but kept my body language sufficiently unfriendly to deter even the loneliest Westerner from attempting to strike up a conversation. If foreigners were unremarkable in most of the city, they were positively eye-glazing here
even to each other
and I did my best to offer no one the slightest reason to remember me.
Along the way I checked for messages from Alison, but there were none. I left five of my own, tiny abstract chalk marks on bus shelters and park benches
all slightly different, but all saying the same thing: CLOSE BRUSH, BUT SAFE NOW. MOVING ON.
By early evening, I'd done all I could to throw off my hypothetical shadow, so I headed for the next hotel on our agreed but unwritten list. The last time we'd met face-to-face, in Hanoi, I'd mocked all of Alison's elaborate preparations. Now I was beginning to wish that I'd begged her to extend our secret language to cover more extreme contingencies. FATALLY WOUNDED. BETRAYED YOU UNDER TORTURE. REALITY DECAYING. OTHERWISE FINE.
The hotel on Huaihai Zhonglu was a step up from the last one, but not quite classy enough to refuse payment in cash. The desk clerk made polite small-talk, and I lied as smoothly as I could about my plans to spend a week sight-seeing before heading for Beijing. The bellperson smirked when I tipped him too much
and I sat on my bed for five minutes afterward, wondering what significance to read into
I struggled to regain a sense of proportion. Industrial Algebra
have bribed every single hotel employee in Shanghai to be on the lockout for us
but that was a bit like saying that, in theory, they could have duplicated our entire twelve-year search for defects, and not bothered to pursue us at all. There was no question that they wanted what we had, badly
but what could they act
ally do about it? Go to a merchant bank (or the Mafia, or a Triad) for finance? That might have worked if the cargo had been a stray kilogram ofplutonium, or a valuable gene sequence
but only a few hundred thousand people on the planet would be capable of understanding what the defect
even in theory. Only a fraction of that number would believe that such a thing could really exist
. . .
and even fewer would be both wealthy and immoral enough to invest in the bus
ness of exploiting it.
The stakes appeared to be infinitely high
but that didn't make the players omnipotent. Not yet.
I changed the dressing on my arm, from sock to handkerchief, but the incision was deeper than I'd realized, and it was still bleeding thinly. I left the hotel
and found exactly what I needed in a twenty-four-hour emporium just ten minutes away. Surgical grade tissue repair cream: a mixture of collagen-based adhesive, antiseptic, and growth factors. The emporium wasn't even a pha
it just had aisle after aisle packed with all kinds of unrelated odds and ends, laid out beneath the unblinking blue-white ceiling panels. Canned food, PVC plumbing fixtures, traditional medicines, rat contraceptives, video ROMS. It was a random cornucopia, an almost organic diversity
as if the products had all just grown on the shelves from whatever spores the wind had happened to blow in.