by Greg Egan
There was no difference between the model of Yatima’s beliefs about the other citizens, buried inside the symbol for Yatima ... and the models of the other citizens themselves, inside their respective symbols. The network finally recognized this, and began to discard the unnecessary intermediate stages. The model for Yatima’s beliefs became the whole, wider network of the orphan’s symbolic knowledge.
And the model of Yatima’s beliefs about Yatima’s mind became the whole model of Yatima’s mind: not a tiny duplicate, or a crude summary, just a tight bundle of connections looping back out to the thing itself.
The orphan’s stream of consciousness surged through the new connections, momentarily unstable with feedback: I think that Yatima thinks that I think that Yatima thinks ...
Then the symbol network identified the last redundancies, cut a few internal links, and the infinite regress collapsed into a simple, stable resonance:
I am thinking —
I am thinking that I know what I’m thinking.
Yatima said, “I know what I’m thinking.”
Inoshiro replied airily, “What makes you think anyone cares?”
For the five-thousand-and-twenty-third time, the conceptory checked the architecture of the orphan’s mind against the polis’s definition of self-awareness.
Every criterion was now satisfied.
The conceptory reached into the part of itself which ran the womb, and halted it, halting the orphan. It modified the machinery of the womb slightly, allowing it to run independently, allowing it to be reprogrammed from within. Then it constructed a signature for the new citizen — two unique megadigit numbers, one private, one public — and embedded them in the orphan’s cypherclerk, a small structure which had lain dormant, waiting for these keys. It sent a copy of the public signature out into the polis, to be catalogued, to be counted.
Finally, the conceptory passed the virtual machine which had once been the womb into the hands of the polis operating system, surrendering all power over its contents. Cutting it loose, like a cradle set adrift in a stream. It was now the new citizen’s exoself: its shell, its non-sentient carapace. The citizen was free to reprogram it at will, but the polis would permit no other software to touch it. The cradle was unsinkable, except from within.
Inoshiro said, “Stop it! Who are you pretending to be now?”
Yatima didn’t need to part the navigators; ve knew vis icon hadn’t changed appearance, but was now sending out a gestalt tag. It was the kind ve’d noticed the citizens broadcasting the first time ve’d visited the flying-pig scape.
Blanca sent Yatima a different kind of tag; it contained a random number encoded via the public half of Yatima’s signature. Before Yatima could even wonder about the meaning of the tag, vis cypherclerk responded to the challenge automatically: decoding Blanca’s message, re-encrypting it via Blanca’s own public signature, and echoing it back as a third kind of tag. Claim of identity. Challenge. Response.
Blanca said, “Welcome to Konishi, Citizen Yatima.” Ve turned to Inoshiro, who repeated Blanca’s challenge then muttered sullenly, “Welcome, Yatima.”
Gabriel said, “And Welcome to the Coalition of Polises.”
Yatima gazed at the three of them, bemused — oblivious to the ceremonial words, trying to understand what had changed inside verself. Ve saw vis friends, and the stars, and the crowd, and sensed vis own icon ... but even as these ordinary thoughts and perceptions flowed on unimpeded, a new kind of question seemed to spin through the black space behind them all. Who is thinking this? Who is seeing these stars, and these citizens? Who is wondering about these thoughts, and these sights?
And the reply came back, not just in words, but in the answering hum of the one symbol among the thousands that reached out to claim all the rest. Not to mirror every thought, but to bind them. To hold them together, like skin.
Who is thinking this?
I am.
* * *
2
–
Truth Mining
« ^ »
Konishi polis, Earth
23 387 281 042 016 CST
18 May 2975, 10:10:39.170 UT
“What is it you’re having trouble with?” Radiya’s icon was a fleshless skeleton made of twigs and branches, the skull carved from a knotted stump. Vis homescape was a forest of oak; they always met in the same clearing. Yatima wasn’t sure if Radiya spent much time here, or whether ve immersed verself completely in abstract mathematical spaces whenever ve was working, but the forest’s complex, arbitrary messiness made a curiously harmonious backdrop for the spartan objects they conjured up to explore.
“Spatial curvature. I still don’t understand where it comes from.” Yatima created a translucent blob, floating between ver and Radiya at chest height, with half a dozen black triangles embedded in it. “If you start out with a manifold, shouldn’t you be able to impose any geometry you like on it?” A manifold was a space with nothing but dimension and topology; no angles, no distances, no parallel lines. As ve spoke, the blob stretched and bent, and the sides of the triangles swayed and undulated. “I thought curvature existed on a whole new level, a new set of rules you could write any way you liked. So you could choose zero curvature everywhere, if that’s what you wanted.” Ve straightened all the triangles into rigid, planar figures. “Now I’m not so sure. There are some simple two-dimensional manifolds, like a sphere, where I can’t see how to flatten the geometry. But I can’t prove that it’s impossible, either.”
Radiya said, “What about a torus? Can you give a torus Euclidean geometry?”
“I couldn’t at first. But then I found a way.”
“Show me.”
Yatima banished the blob and created a torus, one delta wide and a quarter of a delta high, its white surface gridded with red meridians and blue circles of latitude. Ve’d found a standard tool in the library for treating the surface of any object as a scape; it re-scaled everything appropriately, forced notional light rays to follow the surface’s geodesics, and added a slight thickness so there was no need to become two-dimensional yourself. Politely offering the address so Radiya could follow, Yatima jumped into the torus’s scape.
They arrived standing on the outer rim — the torus’s “equator” — facing “south.” With light rays clinging to the surface the scape appeared boundless, though Yatima could clearly see the backs of both Radiya’s icon and vis own, one short revolution ahead, and ve could just make out a twice-distant Radiya through the gap between the two of them. The forest clearing was nowhere to be seen; above them was nothing but blackness.
Looking due south the perspective was very nearly linear, with the red meridians wrapping the torus appearing to converge toward a distant vanishing point. But to the east and west the blue lines of latitude — which seemed almost straight and parallel nearby — appeared to veer apart wildly as they approached a critical distance. Light rays circumnavigating the torus around the outer rim reconverged, as if focused by a magnifying lens, at the point directly opposite the place where they started out — so the vastly distended image of one tiny spot on the equator, exactly halfway around the torus, was hogging the view and pushing aside the image of everything north or south of it. Beyond the halfway mark the blue lines came together again and exhibited something like normal perspective for a while, before they came full circle and the effect was repeated. But this time the view beyond was blocked by a wide band of purple with a thin rim of black on top, stretching across the horizon: Yatima’s own icon, distorted by the curvature. A green and brown streak was also visible, partly obscuring the purple and black one, if Yatima looked directly away from Radiya.
“The geometry of this embedding is non-Euclidean, obviously.” Yatima sketched a few triangles on the surface at their feet. “The sum of the angles of a triangle depends on where you put it: more than 180 degrees here, near the outer rim, but less than 180 near the inner rim. In between, it almost balances out.”
Radiya nodded. “All right.
So how do you balance it out everywhere — without changing the topology?”
Yatima sent a stream of tags to the scape object, and the view around them began to be transformed. Their smeared icons on the horizon to the east and west began to shrink, and the blue lines of latitude began to straighten out. To the south, the narrow region of linear perspective was expanding rapidly. “If you bend a cylinder into a torus, the lines parallel to the cylinder’s axis get stretched into different-sized circles; that’s where the curvature really comes from. And if you tried to keep all those circles the same size, there’d be no way to keep them apart; you’d crush the cylinder flat in the process. But that’s only true in three dimensions.”
The grid lines were all straight now, the perspective perfectly linear everywhere. They appeared to be standing on a boundless plane, with only the repeated images of their icons to reveal otherwise. The triangles had straightened out, too; Yatima made two identical copies of one of them, then maneuvered the three together into a fan that showed the angles summing to 180 degrees. “Topologically, nothing’s changed; I haven’t made any cuts or joins in the surface. The only difference is ...”
Ve jumped back to the forest clearing. The torus appeared to have been transformed into a short cylindrical band; the large blue circles of latitude were all of equal size now — but the smaller red circles, the meridians, looked like they’d been flattened into straight lines. “I rotated each meridian 90 degrees, into a fourth spatial dimension. They only look flat because we’re seeing them edge-on.” Yatima had rehearsed the trick with a lower-dimensional analogue: taking the band between a pair of concentric circles and twisting it 90 degrees out of the plane, standing it up on its edge; the extra dimension created room for the entire band to have a uniform radius. With a torus it was much the same; every circle of latitude could have the same radius, so long as they were given different “heights” in a fourth dimension to keep them apart.
Yatima re-colored the whole torus in smoothly varying shades of green to reveal the hidden fourth coordinate. The inner and outer surfaces of the “cylinder” only matched colors at the top and bottom rims, where they met up in the fourth dimension; elsewhere, different hues on either side showed that they remained separated.
Radiya said, “Very nice. Now can you do the same for a sphere?”
Yatima grimaced with frustration. “I’ve tried! Intuitively, it just looks impossible ... but I would have said the same thing about the torus, before I found the right trick.” Ve created a sphere as ve spoke, then deformed it into a cube. No good, though — that was just sweeping all the curvature into the singularities of the corners, it didn’t make it go away.
“Okay. Here’s a hint.” Radiya turned the cube back into a sphere, and drew three great circles on it in black: an equator, and two complete meridians 90 degrees apart.
“What have I divided the surface into?”
“Triangles. Eight triangles.” Four in the northern hemisphere, four in the south.
“And whatever you do to the surface — bend it, stretch it, twist it into a thousand other dimensions — you’ll always be able to divide it up the same way, won’t you? Eight triangles, drawn between six points?”
Yatima experimented, deforming the sphere into a succession of different shapes. “I think you’re right. But how does that help?”
Radiya remained silent. Yatima made the object transparent, so ve could see all the triangles at once. They formed a kind of coarse mesh, a six-pointed net, a closed bag of string. Ve straightened all twelve lines, which certainly flattened the triangles — but it transformed the sphere into an octahedral diamond, which was just as bad as a cube. Each face of the diamond was perfectly Euclidean, but the six sharp points were like infinitely concentrated repositories of curvature.
Ve tried smoothing and flattening the six points. That was easy — but it made the eight triangles as bowed and non-Euclidean as they’d been on the original sphere. It seemed “obvious” that the points and the triangles could never be made flat simultaneously ... but Yatima still couldn’t pin down the reason why the two goals were irreconcilable. Ve measured the angles where four triangles met, around what had once been a point of the diamond: 90, 90, 90, 90. That much made perfect sense: to lie flat, and meet nicely without any gaps, they had to add up to 360 degrees. Ve reverted to the unblunted diamond, and measured the same angles again: 60, 60, 60, 60. A total of 240 was too small to lie flat; anything less than a full circle forced the surface to roll up like the point of a cone ...
That was it! That was the heart of the contradiction! Every vertex needed angles totaling 360 degrees around it, in order to lie flat... while every flat, Euclidean triangle supplied just 180 degrees. Half as much. So if there’d been exactly twice as many triangles as vertices, everything would have added up perfectly — but with six vertices and only eight triangles, there wasn’t enough flatness to go round.
Yatima grinned triumphantly, and recounted vis chain of reasoning. Radiya said calmly, “Good. You’ve just discovered the Gauss-Bonnet Theorem, linking the Euler number and total curvature.”
“Really?” Yatima felt a surge of pride; Euler and Gauss were legendary miners — long-dead fleshers, but their skills had rarely been equaled.
“Not quite.” Radiya smiled slightly. “You should look up the precise statement of it, though; 1 think you’re ready for a formal treatment of Riemannian spaces. But if it all starts to seem too abstract, don’t be afraid to back off and play around with some more examples.”
“Okay.” Yatima didn’t need to be told that the lesson was over. Ve raised a hand in a gesture of thanks, then withdrew vis icon and viewpoint from the clearing.
For a moment Yatima was scapeless, input channels isolated, alone with vis thoughts. Ve knew ve still didn’t understand curvature fully — there were dozens of other ways to think about it — but at least ve’d grasped one more fragment of the whole picture. Then ve jumped to the Truth Mines.
Ve arrived in a cavernous space with walls of dark rock, aggregates of gray igneous minerals, drab brown clays, streaks of rust red. Embedded in the floor of the cavern was a strange, luminous object: dozens of floating sparks of light, enclosed in an elaborate set of ethereal membranes. The membranes formed nested, concentric families, Daliesque onion layers — «ach series culminating in a bubble around a single spark, or occasionally a group of two or three. As the sparks drifted, the membranes flowed to accommodate them, in such a way that no spark ever escaped a single level of enclosure.
In one sense, the Truth Mines were just another indexscape. Hundreds of thousands of specialized selections of the library’s contents were accessible in similar ways — and Yatima had climbed the Evolutionary Tree, hopscotched the Periodic Table, walked the avenue-like Timelines for the histories of fleshers, gleisners, and citizens. Half a megatau before, ve’d swum through the Eukaryotic Cell; every protein, every nucleotide, every carbohydrate drifting through the cytoplasm had broadcast gestalt tags with references to everything the library had to say about the molecule in question.
In the Truth Mines, though, the tags weren’t just references; they included complete statements of the particular definitions, axioms, or theorems the objects represented. The Mines were self-contained: every mathematical result that fleshers and their descendants had ever proven was on display in its entirety. The library’s exegesis was helpful — but the truths themselves were all here.
The luminous object buried in the cavern floor broadcast the definition of a topological space: a set of points (the sparks), grouped into “open subsets” (the contents of one or more of the membranes) which specified how the points were connected to each other — without appealing to notions like “distance” or “dimension.” Short of a raw set with no structure at all, this was about as basic as you could get: the common ancestor of virtually every entity worthy of the name “space,” however exotic. A single tunnel led into the cavern, providing a link to the necessary prior concepts
, and half a dozen tunnels led out, slanting gently “down” into the bedrock, pursuing various implications of the definition. Suppose T is a topological space ... then what follows? These routes were paved with small gemstones, each one broadcasting an intermediate result on the way to a theorem.
Every tunnel in the Mines was built from the steps of a watertight proof; every theorem, however deeply buried, could be traced back to every one of its assumptions. And to pin down exactly what was meant by a “proof,” every field of mathematics used its own collection of formal systems: sets of axioms, definitions, and rules of deduction, along with the specialized vocabulary needed to state theorems and conjectures precisely.
When ve’d first met Radiya in the Mines, Yatima had asked ver why some non-sentient program couldn’t just take each formal system used by the miners and crank out all its theorems automatically — sparing citizens the effort.
Radiya had replied, “Two is prime. Three is prime. Five is prime. Seven is prime. Eleven is prime. Thirteen is prime. Seventeen is —”
“Stop!”
“If I didn’t get bored, I could go on like that until the Big Crunch, and discover nothing else.”
“But we could run a few billion programs at once, all mining in different directions. It wouldn’t matter if some of them never found anything interesting.”
“Which ‘different directions’ would you choose?”
“I don’t know. All of them?”
“A few billion blind moles won’t let you do that. Suppose you have just one axiom, taken as given, and ten valid logical steps you can use to generate new statements. After one step, you have ten truths to explore.” Radiya had demonstrated, building a miniature, branching mine in the space in front of Yatima. “After ten steps, you have ten billion, ten to the tenth power.” The fan of tunnels in the toy mine was already an unresolvable smear — but Radiya filled them with ten billion luminous moles, making the coal face glow strongly. “After twenty steps, you have ten to the twentieth. Too many to explore at once, by a factor of ten billion. How are you going to choose the right ones? Or would you time-share the moles between all of these paths — slowing them down to the point of uselessness?” The moles spread their light out proportionately — and the glow of activity became invisibly feeble. “Exponential growth is a curse in all its forms. You know it almost wiped out the fleshers? If we were insane enough, we could try turning the whole planet — or the whole galaxy — into some kind of machine able to exert the necessary brute computational force ... but even then, I doubt we’d reach Fermat’s Last Theorem before the end of the universe.”