by Greg Egan
“And this goes on generation after generation? Like some secret society? The Knights of the Arithmetic Inconsistency?”
“I’ll work on the crest.”
We 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.”
“OK. Good luck.” Her eyelids were starting to droop.
“Take care of yourself.”
Alison 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 yourself?”
“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.”
“Yeah, 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 the net.”
Her frown deepened. “When will you be back?”
“I’m 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 to that.”
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 time together.
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 OK?”
“It’s 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 have bothered.
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 pin-prick.
#
Victoria 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 PhD and a trickle of publications, however tenuously related to the topic of the seminar, were still enough to make me welcome.
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.
I sat 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.
“Dr. Campbell?”
He 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.”
“Of 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?
“I 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.”
As 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.
“The 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.”
“Were you one of our volunteers?” I’d checked the records, and he wasn’t listed, but people had had the option of registering anonymously.
“No. 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 naïveté of our hypothesis. Before things turned serious, it was the entertainment value of that backlash that had made our efforts worthwhile.
“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.
He said, “What you and Dr. Tierney s
uggested 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 summary?”
“Sure.”
“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 14 billion years ago.”
I 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 over.
I said, “And yet you still believe it’s worth searching for an inconsistency?”
“I 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 us.
“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 scone’.”
“Right.”
“You know how banks count large quantities of cash?”
“By 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 with nit-picking.
“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 integer addition.”
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.
“He 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 of matter.”
“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 reality.
I said, “It’s an interesting idea, but I still don’t see how it could speed up the hunt for inconsistencies.”
“I 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.”
“But 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.”
“Taking 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.
“That’s 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.
Was 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.
Campbell 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.”
“I 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 witho
ut 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?
Campbell 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.”
#
Campbell’s 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.
Campbell’s 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 for long.
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 original OS.
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.