Pirit had insisted that only Sando, the senior archaeologist, be told about Joan's true nature; Joan refused to lie to anyone, but had agreed to tell her colleagues only that she was a mathematician and that she was not permitted to discuss her past. At first this had made them guarded and resentful, no doubt because they assumed that she was some kind of spy sent by the authorities to watch over them. Later it had dawned on them that she was genuinely interested in their work, and that the absurd restrictions on her topics of conversation were not of her own choosing. Nothing about the Noudah's language or appearance correlated strongly with their recent division into nations—with no oceans to cross, and a long history of migration they were more or less geographically homogeneous—but Joan's odd name and occasional faux pas could still be ascribed to some mysterious exoticism. Rali and Surat seemed content to assume that she was a defector from one of the smaller nations, and that her history could not be made explicit for obscure political reasons.
"There are more tablets here, very close to the surface," Rali announced excitedly. "The acoustics are unmistakable." Ideally they would have excavated the entire hillside, but they did not have the time or the labour, so they were using acoustic tomography to identify likely deposits of accessible Niah writing, and then concentrating their efforts on those spots.
The Niah had probably had several ephemeral forms of written communication, but when they found something worth publishing, it stayed published: they carved their symbols into a ceramic that made diamond seem like tissue paper. It was almost unheard of for the tablets to be broken, but they were small, and multitablet works were sometimes widely dispersed. Niah technology could probably have carved three million years' worth of knowledge onto the head of a pin—they seemed not to have invented nanomachines, but they were into high-quality bulk materials and precision engineering—but for whatever reason they had chosen legibility to the naked eye above other considerations.
Joan made herself useful, taking acoustic readings farther along the slope, while Sando watched over his students as they came closer to the buried Niah artifacts. She had learnt not to hover around expectantly when a discovery was imminent; she was treated far more warmly if she waited to be summoned. The tomography unit was almost foolproof, using satellite navigation to track its position and software to analyse the signals it gathered; all it really needed was someone to drag it along the rock face at a suitable pace.
From the corner of her eye, Joan noticed her shadow on the rocks flicker and grow complicated. She looked up to see three dazzling beads of light flying west out of the sun. She might have assumed that the fusion ships were doing something useful, but the media was full of talk of "military exercises," which meant the Tirans and the Ghahari were engaging in expensive, belligerent gestures in orbit, trying to convince each other of their superior skills, technology, or sheer strength of numbers. For people with no real differences apart from a few centuries of recent history, they could puff up their minor political disputes into matters of the utmost solemnity. It might almost have been funny, if the idiots hadn't incinerated hundreds of thousands of each other's citizens every few decades, not to mention playing callous and often deadly games with the lives of the inhabitants of smaller nations.
"Jown! Jown! Come and look at this!" Surat called to her. Joan switched off the tomography unit and jogged towards the archaeologists, suddenly conscious of her body's strangeness. Her legs were stumpy but strong, and her balance as she ran came not from arms and shoulders but from the swish of her muscular tail.
"It's a significant mathematical result," Rali informed her proudly when she reached them. He'd pressure-washed the sandstone away from the near-indestructible ceramic of the tablet, and it was only a matter of holding the surface at the right angle to the light to see the etched writing stand out as crisply and starkly as it would have a million years before.
Rali was not a mathematician, and he was not offering his own opinion on the theorem the tablet stated; the Niah themselves had had a clear set of typographical conventions which they used to distinguish between everything from minor lemmas to the most celebrated theorems. The size and decorations of the symbols labelling the theorem attested to its value in the Niah's eyes.
Joan read the theorem carefully. The proof was not included on the same tablet, but the Niah had a way of expressing their results that made you believe them as soon as you read them; in this case the definitions of the terms needed to state the theorem were so beautifully chosen that the result seemed almost inevitable.
The theorem itself was expressed as a commuting hypercube, one of the Niah's favourite forms. You could think of a square with four different sets of mathematical objects associated with each of its corners, and a way of mapping one set into another associated with each edge of the square. If the maps commuted, then going across the top of the square, then down, had exactly the same effect as going down the left edge of the square, then across: either way, you mapped each element from the top-left set into the same element of the bottom-right set. A similar kind of result might hold for sets and maps that could naturally be placed at the corners and edges of a cube, or a hypercube of any dimension. It was also possible for the square faces in these structures to stand for relationships that held between the maps between sets, and for cubes to describe relationships between those relationships, and so on.
That a theorem took this form didn't guarantee its importance; it was easy to cook up trivial examples of sets and maps that commuted. The Niah didn't carve trivia into their timeless ceramic, though, and this theorem was no exception. The seven-dimensional commuting hypercube established a dazzlingly elegant correspondence between seven distinct, major branches of Niah mathematics, intertwining their most important concepts into a unified whole. It was a result Joan had never seen before: no mathematician anywhere in the Amalgam, or in any ancestral culture she had studied, had reached the same insight.
She explained as much of this as she could to the three archaeologists; they couldn't take in all the details, but their faces became orange with fascination when she sketched what she thought the result would have meant to the Niah themselves.
"This isn't quite the Big Crunch," she joked, "but it must have made them think they were getting closer." The 'Big Crunch' was her nickname for the mythical result that the Niah had aspired to reach: a unification of every field of mathematics that they considered significant. To find such a thing would not have meant the end of mathematics—it would not have subsumed every last conceivable, interesting mathematical truth—but it would certainly have marked a point of closure for the Niah's own style of investigation.
"I'm sure they found it," Surat insisted. "They reached the Big Crunch, then they had nothing more to live for."
Rali was scathing. "So the whole culture committed collective suicide?"
"Not actively, no," Surat replied. "But it was the search that had kept them going."
"Entire cultures don't lose the will to live," Rali said. "They get wiped out by external forces: disease, invasion, changes in climate."
"The Niah survived for three million years," Surat countered. "They had the means to weather all of those forces. Unless they were wiped out by alien invaders with vastly superior technology." She turned to Joan. "What do you think?"
"About aliens destroying the Niah?"
"I was joking about the aliens. But what about the mathematics? What if they found the Big Crunch?"
"There's more to life than mathematics," Joan said. "But not much more."
Sando said, "And there's more to this find than one tablet. If we get back to work, we might have the proof in our hands before sunset."
5
Joan briefed Halzoun by video link while Sando prepared the evening meal. Halzoun was the mathematician Pirit had appointed to supervise her, but apparently his day job was far too important to allow him to travel. Joan was grateful; Halzoun was the most tedious Noudah she had encountered. He could understand
the Niah's work when she explained it to him, but he seemed to have no interest in it for its own sake. He spent most of their conversations trying to catch her out in some deception or contradiction, and the rest pressing her to imagine military or commercial applications of the Niah's gloriously useless insights. Sometimes she played along with this infantile fantasy, hinting at potential superweapons based on exotic physics that might come tumbling out of the vacuum, if only one possessed the right Niah theorems to coax them into existence.
Sando was her minder too, but at least he was more subtle about it. Pirit had insisted that she stay in his shelter, rather than sharing Rali and Surat's; Joan didn't mind, because with Sando she didn't have the stress of having to keep quiet about everything. Privacy and modesty were nonissues for the Noudah, and Joan had become Noudah enough not to care herself. Nor was there any danger of their proximity leading to a sexual bond; the Noudah had a complex system of biochemical cues that meant desire only arose in couples with a suitable mixture of genetic differences and similarities. She would have had to search a crowded Noudah city for a week to find someone to lust after, though at least it would have been guaranteed to be mutual.
After they'd eaten, Sando said, "You should be happy. That was our best find yet."
"I am happy." Joan made a conscious effort to exhibit a viridian tinge. "It was the first new result I've seen on this planet. It was the reason I came here, the reason I travelled so far."
"Something's wrong, though, I think."
"I wish I could have shared the news with my friend," Joan admitted. Pirit claimed to be negotiating with the Tirans to allow Anne to communicate with her, but Joan was not convinced that she was genuinely trying. She was sure that Pirit would have relished the thought of listening in on a conversation between the two of them—while forcing them to speak Noudah, of course—in the hope that they'd slip up and reveal something useful, but at the same time she would have had to face the fact that the Tirans would be listening too. What an excruciating dilemma.
"You should have brought a communications link with you," Sando suggested. "A home-style one, I mean. Nothing we could eavesdrop on."
"We couldn't do that," Joan said.
He pondered this. "You really are afraid of us, aren't you? You think the smallest technological trinket will be enough to send us straight to the stars, and then you'll have a horde of rampaging barbarians to deal with."
"We know how to deal with barbarians," Joan said coolly.
Sando's face grew dark with mirth. "Now I'm afraid."
"I just wish I knew what was happening to her," Joan said. "What she was doing, how they were treating her."
"Probably much the same as we're treating you," Sando suggested. "We're really not that different." He thought for a moment. "There was something I wanted to show you." He brought over his portable console, and summoned up an article from a Tiran journal. "See what a borderless world we live in," he joked.
The article was entitled "Seekers and Spreaders: What We Must Learn from the Niah." Sando said, "This might give you some idea of how they're thinking over there. Jaqad is an academic archaeologist, but she's also very close to the people in power."
Joan read from the console while Sando made repairs to their shelter, secreting a molasses-like substance from a gland at the tip of his tail and spreading it over the cracks in the walls.
There were two main routes a culture could take, Jaqad argued, once it satisfied its basic material needs. One was to think and study: to stand back and observe, to seek knowledge and insight from the world around it. The other was to invest its energy in entrenching its good fortune.
The Niah had learnt a great deal in three million years, but in the end it had not been enough to save them. Exactly what had killed them was still a matter of speculation, but it was hard to believe that if they had colonised other worlds they would have vanished on all of them. "Had the Niah been Spreaders," Jaqad wrote, "we might expect a visit from them, or them from us, sometime in the coming centuries."
The Noudah, in contrast, were determined Spreaders. Once they had the means, they would plant colonies across the galaxy. They would, Jaqad was sure, create new biospheres, reengineer stars, and even alter space and time to guarantee their survival. The growth of their empire would come first; any knowledge that failed to serve that purpose would be a mere distraction. "In any competition between Seekers and Spreaders, it is a Law of History that the Spreaders must win out in the end. Seekers, such as the Niah, might hog resources and block the way, but in the long run their own nature will be their downfall."
Joan stopped reading. "When you look out into the galaxy with your telescopes," she asked Sando, "how many reengineered stars do you see?"
"Would we recognise them?"
"Yes. Natural stellar processes aren't that complicated; your scientists already know everything there is to know about the subject."
"I'll take your word for that. So . . . you're saying Jaqad is wrong? The Niah themselves never left this world, but the galaxy already belongs to creatures more like them than like us?"
"It's not Noudah versus Niah," Joan said. "It's a matter of how a culture's perspective changes with time. Once a species conquers disease, modifies their biology, and spreads even a short distance beyond their homeworld, they usually start to relax a bit. The territorial imperative isn't some timeless Law of History; it belongs to a certain phase."
"What if it persists, though? Into a later phase?"
"That can cause friction," Joan admitted.
"Nevertheless, no Spreaders have conquered the galaxy?"
"Not yet."
Sando went back to his repairs; Joan read the rest of the article. She'd thought she'd already grasped the lesson demanded by the subtitle, but it turned out that Jaqad had something more specific in mind.
"Having argued this way, how can I defend my own field of study from the very same charges as I have brought against the Niah? Having grasped the essential character of this doomed race, why should we waste our time and resources studying them further?
"The answer is simple. We still do not know exactly how and why the Niah died, but when we do, that could turn out to be the most important discovery in history. When we finally leave our world behind, we should not expect to find only other Spreaders to compete with us, as honourable opponents in battle. There will be Seekers as well, blocking the way: tired, old races squatting uselessly on their hoards of knowledge and wealth.
"Time will defeat them in the end, but we already waited three million years to be born; we should have no patience to wait again. If we can learn how the Niah died, that will be our key, that will be our weapon. If we know the Seekers' weakness, we can find a way to hasten their demise."
6
The proof of the Niah's theorem turned out to be buried deep in the hillside, but over the following days they extracted it all.
It was as beautiful and satisfying as Joan could have wished, merging six earlier, simpler theorems while extending the techniques used in their proofs. She could even see hints of how the same methods might be stretched further to yield still stronger results. The "Big Crunch" had always been a slightly mocking, irreverent term, but now she was struck anew by how little justice it did to the real trend that had fascinated the Niah. It was not a matter of everything in mathematics collapsing in on itself, with one branch turning out to have been merely a recapitulation of another under a different guise. Rather, the principle was that every sufficiently beautiful mathematical system was rich enough to mirror in part—and sometimes in a complex and distorted fashion—every other sufficiently beautiful system. Nothing became sterile and redundant, nothing proved to have been a waste of time, but everything was shown to be magnificently intertwined.
After briefing Halzoun, Joan used the satellite dish to transmit the theorem and its proof to the decoy node. That had been the deal with Pirit: anything she learnt from the Niah belonged to the whole galaxy, as long as she explain
ed it to her hosts first.
The archaeologists moved across the hillside, hunting for more artifacts in the same layer of sediment. Joan was eager to see what else the same group of Niah might have published. One possible eight-dimensional hypercube was hovering in her mind; if she'd sat down and thought about it for a few decades she might have worked out the details herself, but the Niah did what they did so well that it would have seemed crass to try to follow clumsily in their footsteps when their own immaculately polished results might simply be lying in the ground, waiting to be uncovered.
A month after the discovery, Joan was woken by the sound of an intruder moving through the shelter. She knew it wasn't Sando; even as she slept an ancient part of her Noudah brain was listening to his heartbeat. The stranger's heart was too quiet to hear, which required great discipline, but the shelter's flexible adhesive made the floor emit a characteristic squeak beneath even the gentlest footsteps. As she rose from her couch she heard Sando waking, and she turned in his direction.
Bright torchlight on his face dazzled her for a moment. The intruder held two knives to Sando's respiratory membranes; a deep enough cut there would mean choking to death, in excruciating pain. The nanomachines that had built Joan's body had wired extensive skills in unarmed combat into her brain, and one scenario involving a feigned escape attempt followed by a sideways flick of her powerful tail was already playing out in the back of her mind, but as yet she could see no way to guarantee that Sando came through it all unharmed.
She said, "What do you want?"
The intruder remained in darkness. "Tell me about the ship that brought you to Baneth."
"Why?"
"Because it would be a shame to shred your colleague here, just when his work was going so well." Sando refused to show any emotion on his face, but the blank pallor itself was as stark an expression of fear as anything Joan could imagine.
The Best Science Fiction and Fantasy of the Year-II Page 13