As soon as this thought hardened, the number–storm intensified. A tide of ideoplasts began to build and flow and rage before my inner sight. I was excited almost beyond control. For the thousandth time I contemplated the deceptively simple statement of the Hypothesis: that between any pair of discrete Lavi sets of point–sources there exists a one–to–one mapping. I broke the statement apart and examined the pieces. What, exactly, was a Lavi set? What was a point–source? Was I sure I understood the difference between a Lavi set and a discrete Lavi set? How could I show the mapping was one–to–one, and more importantly, how might I construct the mapping to begin with? At first I fell into old thoughtways and rediscovered my old attempts to find a solution. Often I found myself reasoning in circles. I grew discouraged at the shallowness of my thinking. How could I prove this? How could I prove that? How could I break the rusted chain of my habitual, uninspired thoughts?
I tried to restate the problem in a different form, hoping that a new way of looking at it might enable me to see the obvious. And though I did find an equivalent statement, it proved even more opaque than the original. I decomposed the Hypothesis, recombined it into a slightly different statement—all to no avail. In my mind, I represented the pieces of the Hypothesis with pictures in order to “see” relations that I might have overlooked. I generalized the Hypothesis to include all Lavi sets, and I played with mappings of specific Lavi sets which were quite well known; I tried proof by contradiction; I dissected related theorems (Bardo’s Boomerang Theorem, incidentally, is closely related, though simpler to prove); I followed long, dark corridors of reasoning down thousands of steps; I cursed and I rubbed my eyes and temples, and finally, when my beard and hair were rimed with crusts of dried sweat and I had nearly abandoned hope, I began to make wild guesses.
I do not know how long I tried to prove the Hypothesis. Days, seconds, years—what did time matter? And yet it did matter. At any given time, Soli might be close to his moment of inspiration. The race went on, and measureless moments passed into endless days, and I began to think the Hypothesis was unprovable. For a long while I tried to show that it was unprovable, even though I did not really believe it could be so. My intuition—and a mathematical man should never ignore his intuition—something within me whispered that the Hypothesis was indeed provable, and more, that the proof would seem embarrassingly obvious once I had found it. If it could be found. If I could find it. If...If one mapping between a pair of discrete Lavi sets of point–sources exists, then there are an infinite number of mappings; if one covers an n–dimensional cube with finitely many sufficiently small closed sets, then there are necessarily points which belong to at least n+1 of these sets; if one stirs a bowl of blood tea for a thousand years, there will exist at least one point—one corpuscle of blood—which will remain fixed in its original position, undisturbed by the stirring; if/then; if I examined the ideoplasts of the Tycho’s Conjecture and the Tiling Theorem and the Fixed–Point Theorem, if I broke down the brilliant crystalline arrays into the simple shards of proof steps instead of clumping the arrays together, then I might better understand the inspirations leading to the proofs of these famous theorems. If I understood the proofs better, then I might better use the theorems to prove the Great Theorem.
If...If a pilot dwelt too long in dreamtime, then he must withdraw from the thoughtspace and sleep. I was suddenly tired of the interplay of ideoplasts flooding my mind; I dreaded thinking ever again a single mathematical thought. I bit my lip and cursed and despaired. Finally, I slept. I closed my eyes and mind to the number– storm, and I floated like a corpse in a deathship. I slept a long time. When I at last awoke my eyelids were stuck together; my mouth tasted of blood. Probably I had ground my teeth while I dreamed. My mind was dark and cold, like black ice. I was as empty as an abandoned snow hut on the shelf of the deep winter sea. And yet the cold was not total. There was a flush of warmth inside me as if I had been rescued from a capsized sled and fed a bowl of hot tea. The faint flame of an idea burned in my mind. Where it had come from I did not know. For no particular reason I thought of an obscure theorem, the Justerini Mapping Theorem. The flame grew brighter, as if I had breathed on the glowing embers of a woodfire. I was very excited. How elegant, I thought, was the collapse of the omega function by which Olaf Justerini had proved his neglected theorem! How beautiful!
I began thinking in a general manner of the whole structure of the Continuum Hypothesis. I saw, in a general, hazy way, how the same idea leading to the collapse of Justerini’s omega function might be applied to collapse the Lavi correspondence scheme. I trembled with excitement and with fear, too, because I had had a thousand such hazy ideas to collapse the correspondence scheme. But this idea was different—I could almost see the difference. Somehow my idea felt right; somehow it fitted and completed the holes in my pattern of thinking. I reached out to my ship’s neurologics, and there was light. Ideoplasts swirled around the still point of my mind, and the manifold opened. I entered again into dreamtime. I had an anticipation of rightness as I translated my idea into the diamond crystal of a new ideoplast. The flames of my thoughts grew hotter. I built up the array of my proof. The Lavi correspondence scheme could be made to collapse if and only if the Justerini subspace was embedded within the simple Lavi space. Could I show it was embedded? It must be embedded; there must be a simple series of steps to show it was embedded; even a novice could show it was embedded. My thoughts burned like lava in my brain. My brain itself felt electric and different, vastly more capacious, as if it could hold an ocean of molten thoughts. I felt that I was thinking in a way I never had before, not even when I faced my computer and hurried my thoughts with slowtime. Now my thoughts came much faster. Whole new concepts came to me and fell into place, all in a flash. I understood things. How can I describe the exquisite pain and pleasure of this understanding, this wonderful vision of orderedness? My thoughts seared me; my thoughts were flaming crimson; my thoughts were burning raindrops of light. The subspace was embedded within the simple Lavi space! And then the correspondence scheme collapsed, almost as the layers of a star collapse around the core when it becomes a supernova, and there was a choosable mapping. There was a cheosable mapping! There was elegance, beauty and starlight. I made a mapping. The white light of dreamtime swept by in brilliant streamers, then collapsed into a single point of starlight which burned and expanded and brightened until it filled all of my mind.
Oh, Soli, I thought, the race goes on, but this race is over.
I fell out in realspace above the hot white star that the Entity had named Gehenna Luz. I had proved the Great Theorem; I had journeyed far at a single fall, and now all the stars in the sky were finally mine.
26
Kalinda of the Flowers
When Man took to his bed the Computer, there was great rejoicing, and great fear, too, for their children were almost like gods. The mainbrains bestrode the galaxy at will, and changed its very face. The Silicon God, The Solid State Entity, AI Squared, Enth Generation—their names are many. And there were the Carked and the Symbionts, whose daughters were the Neurosingers, Warrior–Poets, the Neurologicians and the Pilots of the Order of Mystic Mathematicians. So beautiful were these daughters that man longed to touch them, but touch them he could not. And so was born the Second Law of the Civilized Worlds, which was that Man could not stare too long at the faces of the Computer or her children, and still remain as Man.
from A Requiem for Homo Sapiens, by Horthy Hosthoh
Gehenna Luz was a beautiful star. It was a massive star, white and scintillant and hot. I was ravished by its beauty. Why stars, I wondered? Why was there anything at all? Why do we breathe, suck in sorrow, joy, pity and pain? Why is—
You have proved your theorem, my Pilot, and still you ask these questions.
It was the godvoice of Entity inside my mind, a voice which I had hoped I would never have to hear again. But She had prophesied that I would return to her, and return I had.
The stars are that w
e might glory in their beauty. And we exist to worship light.
I remembered how the Entity loved her riddles and her games, and I thought:
—Do you have such a simple answer for every question, then?
I am here to answer your questions.
—Well, I have a thousand questions. Where’s Bardo? Why, if you could have stopped the battle whenever you wished, why did you let him die? Is he dead? Do you know? No! Do not speak to me...like this. I don’t want to listen to your voice, here, inside. How can I guard the privacy of my mind, then?
Human beings do not really want privacy.
There was silence for a while, and then, inside the pit of my ship, appeared the imago of the Tycho, with his walrus jowls and savage grin. He was so close I could have reached out and plunged my hand into the phased light waves that were his bristly face. When he spoke, real sound waves washed my ears: “Would you rather speak as a human being? Then we will speak thus.”
“Where is Soli? The other pilots? What was the result of the battle?”
The Tycho licked spit from his yellow teeth and said, “You have fallen far, no pilot further. The others are worming their way through the manifold. Only you have proven your theorem; only you will be told the secret. Fix your telescopes on the asteroid cluster twelve degrees above the solar plane.”
I oriented my ship’s telescopes according to his/Her directions. I looked out into the space a billion miles from Gehenna Luz, at a great cloud of asteroids, rocks, dust and other debris. Some of these fragments were huge, cratered and pocked, red with silicates and iron; some were a darker dun color, probably rich with carbon and water compounds. At first I had no idea why the Entity had bade me look at such a bone–heap of pulverized matter. Then, as the ship–computer analysed the carbon, hydrogen, nitrogen and oxygen proportions of one of the smaller asteroids my stomach tightened. I felt an overwhelming apprehension—no, that is not the right word—I immediately knew that something, in a cosmological sense, was very wrong.
“This used to be Gehenna’s single planet,” the Tycho said. “It was twice as massive as Icefall. Now it orbits Gehenna in pieces. Human beings did this. The manswarm tore the planet apart.”
I could hardly believe that She would permit human beings to enter Her brain, to tear planets apart. Then I thought of the decadent human beings I had encountered on my first journey inside the Entity, and I was not so sure.
“How many human beings?” I asked. “Where are these human beings?”
“Fix your telescopes on the long, crescent–shaped asteroid. There, do you see? Look how they shimmer!—their hulls are spun diamond, the same as your lightships.”
I looked through my telescope and saw the terrible image of many man–made worlds. Each world was a spinning cylinder about thirty miles long and ten miles wide. I wondered how many people lived inside each world. I counted the worlds. There were ten thousand four hundred and eight of them. They were like a colony of rod–shaped bacteria spread out against the black blood of space. My first thought was that human beings must have colonized Gehenna Luz before the Entity had grown into this part of the nebula. Perhaps they had even come from Old Earth. A deepship had fallen out, I thought, and the human beings had made a world into which they could increase their numbers. They had mined and smelted and metabolized the elements of the planet to grow habitats and food, to reproduce themselves ten thousand times over. If that was true, they would be among the oldest peoples of the galaxy. (I mean the oldest human peoples.) They must have been here for thousands of years.
When I told this to the Tycho, he pulled his jowls and laughed until the spit ran down his chin. He said, “You know your first thought is wrong. Why don’t you examine your second thought? You must know where these human beings come from.”
“Tell me.”
“Think, Mallory.”
I rubbed my knuckles across my chin and asked, “How long did it take for them to dismantle the planet?”
The Tycho smiled his humorous (or humoring) smile. “Okay, you can calculate how long they’ve been here from the doubling time one of their worlds needs to reproduce itself. It’s an exponential growth. A mathematical man should be able to calculate such things.”
My head hurt, and I pressed my eye and the side of my nose with my fist. I did not know why the Tycho would want to tease me. “What is the doubling time, then? How many years?”
The Tycho’s smile was savage as he said, “Do you mean, how many days?”
“Days!”
“The manswarm breeds quickly, doesn’t it, Pilot? Ten Neverness years ago the first world fell out from the Vild.”
“Ten years ago!”
“They were lost, you see, and they had hungers.”
“Ten years!”
“Shall I show you what human beings can do when they hunger for growth? Are you ready to watch a star explode?”
“Why,” I whispered, “why would they explode their sun? Is it possible?”
I closed my eyes for a moment to watch the telescopic images my ship infused into my visual cortex. I stared at the sifting dust and rocks and ten thousand made–worlds. I wondered again how many people lived in each world. “Mallory,” a voice called out to me. “Mallory, listen.”
I threw my hands over my ears. “No,” I shouted. “The dead don’t have tongues, so they shouldn’t speak.”
I did not want to listen. I did not want to open my eyes. I did not want to hear the dulcet voice or look at the lovely, eyeless face that the Entity was pulling from my memory.
“Oh, Mallory, Mallory!”
At last I could stand it no longer. I opened my eyes and looked at Katharine. She floated before me wearing her white scryer’s robe. Her skin was white as marble, and her eye hollows were deepest black. She smiled at me. “It was foreseen long ago,” she said. “What is has been.”
I wanted to reach out to her, to grab her up in my arms and kiss her full red lips. But I told myself that she was nothing more than light and memory, and dispassionate words hanging in the air. I would not try to touch her, I promised myself. No matter what happened, I would keep my fist pressed against my cheek.
“Why do you torture me, then? Are my crimes so great?” And I cursed and shouted to the Entity, “Bring back the Tycho, damn you! I can talk to him.”
But the Tycho, it seemed, was gone. Katharine’s imago—it was only Katharine’s imago, I reminded myself—answered me: “Long ago the first scryers saw the painful future of...Do you understand the pain of this vision now? Oh, sweet Mallory, with your sweet brain and your sweet life, it hurts more than a man can bear, and so I’ll show what man and woman can...Do you see what I’ve seen? Will you see it if I show you? Look! What has been will be, again and again until all the stars…Do you see?”
In my mind a picture formed. There was a hot white star orbited by a lifeless, frozen planet. Suddenly, from the thickspace close in towards the sun where the photons and radiation spilled out into space like a waterfall of white light, there came the hazy foreimage of an object falling out of the manifold. The image hardened. A clear, diamond cylinder thirty miles long spread its lightsails in a thousand–mile–wide umbrella to capture Gehenna’s profuse radiations. Slowly the light–pressure of trillions of particles against the gossamer, silvery lightsails imparted momentum to the cylinder. It accelerated. In a short time—perhaps only as long as a long deep winter on Neverness—it reached the planet. The cylinder opened. Clouds of tiny disassemblers (or perhaps I should call them programmed bacteria) fell down like a meteor shower through the airless vacuum and gathered over parts of the planet in great patches of shiny dust. Then the disassemblers did their work. They tore loose oxygen atoms from water molecules; they concentrated masses of carbon and other elements. They ate away the very ground. They concentrated hydrogen. In vast reservoirs carved out of the planet’s crust, clouds of hydrogen were stored. Again the cylinder opened, and an army of robot–lasers fell to the planet. They found the hydrogen re
servoirs. Optical crystals in the hearts of the lasers converted infrared light into short wave beams aimed at the hydrogen pocket. The hydrogen grew hot and glowed; it heated to a hundred million degrees and fusioned and exploded. In seconds, great balls of fire and light erupted from the surface. The crust of the planet was vaporized, pulverized, blown into hot dust. Rocks and fused sand fragments were flung out into space. Ice became steam and boiled away. Later, when the dust had settled, the cylinder opened and released still more disassemblers over the chewed–up surface of the planet. In this way, layer by layer, the planet was stripped, crust, mantle and core, torn apart like a dirty snowball and scattered into space.
More pictures formed inside me. I was fascinated with this vision of assembler technology gone wild, so I closed my eyes and watched the disassemblers mine the floating planetary fragments for silicon, mercury and helium, and every other natural element. I watched a cloud of assemblers escape from the cylinder into the newly created asteroids, watched as they bonded carbon atom to atom until they had assembled the gleaming, diamonds hulls of many more cylinders. The assemblers built other things. Telescopes, sulki grids, neurologics, flame globes, shakuhachis, flying wings, chop–sticks, gossilk, trees, houses, glucose pellets, fields of grass—there was no end to the things that the assemblers made. Assemblers made more assemblers, and so the process of converting a planet into ten thousand cylinders did not take very long. Assemblers bonded carbon to hydrogen and oxygen; assemblers had been programmed to fix nitrogen, to build up amino acids and string together proteins. Assemblers could even make human beings. A manswarm of human beings, billions and billions of human beings.
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