Self-Reference Engine

by Toh EnJoe

  So Rita starts by drawing a single line. She thinks about one dimension, and when that becomes problematical, she adds another, always in the direction of additional dimensions. She imagines a point called Rita, somewhere on the right half of the line. On the left half of the line are countless other points.

  QED. It is not necessarily the case that there exists another person almost surely like Rita.

  Rita stops, a smile floating to her lips. This smile is a yellow flag. Once upon a time her grandfather told her this was the smile of the fox pleased with herself at jumping over the pitfall in the path, not noticing the trap that lay beyond.

  Fine. Time to think a little more.

  The point called Rita could be floating all by its lonesome somewhere in space. Fine. But what about all the countless other points just talked about, all squished together on the left half of the line? The point corresponding to Grandpa, for example. Let’s suppose that the point called Grandpa is on the right half of the left half of the line. And all the other countless points are on the left half of the left half of the line. Now we see that for Grandpa too, it is not necessarily the case that there exists another person almost surely like Grandpa. But we still have an infinite number of people on the left half of the left half of the line. My my. Muttering all but wordlessly, with rising intonation, while at the same time raising one eyebrow. My my. Starts to seem we haven’t made much progress at all.

  Forget about Rita and Grandpa. Let’s just think about the same situation for anybody at all: one person, another person. Divide repeatedly: the left half of the left half of the left half of the left half. For each repetition, we can say that on the right half of the right half of the right half, etc., there are no “almost surely like” people, but what about the left half of each of those right halves? There too there are an infinite number of people, each waiting their turn.

  And the left half of the left of half of the (many, many iterations) left half will start to be a very small line segment, in fact, vanishingly small. Still an infinite number of points in that vanishingly small line segment. And the closer the points are to one another, the more closely the people they represent resemble one another.

  In other words, at that end, the number of “almost surely like” people starts to approach infinity.

  Rita tilts her head. What if she were over there? There would be an unlimited, even infinite number of people similar to her. In other words, there would be someone almost surely like her.

  Suddenly, from somewhere in the sky, it came to her: must be universalizable.

  Mark points at random on the line. An infinite number of points. Continue marking as many points as you can, forever, until your arms can’t mark any more, until it starts to look like space itself. No matter how many points you mark, at some distance away from a given point, there will be a spot at which no point is marked. And no matter how far you go from a given point, you will be getting closer to a different point. Maybe that’s what it means.

  Without thinking about it, Rita reaches out one hand, wresting those words away from the sky.

  “An infinite number of people exist who are all ‘almost surely like’ one another.”

  Rita cannot tell if these words are hers or Grandfather’s. Either way, it’s true that this infinite group of almost surely like people exists. And because the number of similar others is infinite, the number of dissimilar others must be finite. One divided by infinity is zero.

  Therefore, if you pick any person at random, the probability that another person “almost surely like” that person does not exist is zero. The same way that rational numbers cannot be depicted on a number line.

  So, the conclusion can only be: “For almost all people, there exist an infinite number of people who are ‘almost surely like’ them.” QED. She still has four days to work out the details.

  Rita does not cry out in exuberance, but she sinks exhausted to the ground, wrapped in a feeling of joy. This is the answer her grandfather was looking for.

  Wrapped in fatigue, Rita thinks, Is Grandfather one of the many people for whom there exists an “almost surely like” person? Is she?

  Or is Grandfather one of those people for whom there can be no duplicate, what with his own brand of fixed ideas and no room for afterthoughts? Or is he the kind who takes some sort of comfort in knowing he is one of the people for whom an “almost surely like” person almost certainly exists?

  No matter how “almost surely like” Grandpa someone else might be, only Grandpa is Grandpa. Even if that other person has memories identical to Grandpa’s, Rita would know her Grandpa.

  And Rita, who is thinking these thoughts, is probably also one of the “most people” of whom an infinite number exist. Rita thinks she would like to meet those Ritas and talk to them. Those people who are “almost surely like” her, who think the same things as her, but who are not her. Those girls are almost certainly somewhere on this unbounded plane. But they may be infinitely far away.

  Right now, in this instant, in her mind Rita is greeting the infinite number of other Ritas who are thinking the same thoughts she is. There is no need to attempt to communicate directly over those long, long distances, so long one might faint at the thought. Those other Ritas are most likely thinking almost exactly the same thing.

  “Hello, you infinite number of people who so closely resemble me, who are ‘almost surely like’ me. Thanks to my grandfather, just today I came to realize that in all likelihood you exist. I am just a little girl. I think you are too. I love my grandfather, and I think you all love your grandfather too. I am very happy to learn that you all almost certainly exist.

  “Just like me, I think almost no one will be interested in listening to you either.

  “Just like me, I think almost no one will understand what you are saying.

  “I am my grandfather’s grandchild.”

  Rita is filled with a feeling of kindness and speaks what springs to her mind. That’s what she said before she came back to herself. I am my grandfather’s grandchild. What was she going to say after that?

  She wonders whether her grandfather realized that there are an infinite number of people in the universe who are almost surely like him and that there is nothing he can do about it. Would he remain detached, deciding that those infinite others had nothing to do with him?

  “Almost surely,” Rita mutters. “With probability one.”

  And then, finally, it dawns on Rita what her grandfather is really trying to say to her.

  That she is a girl who is interesting, worthwhile, funny. Really. Maybe.

  “I am my grandfather’s grandchild. Therefore, I cannot accept that you all may be almost indistinguishably similar to me. Probability one is not identical to certain. I want to be something different. Different from you, different from everyone else.

  “Like plucking a single needle from among the infinite grains of sand. And then throwing that needle away and finding it again. As you have decided. With probability one, you probably all think that what I’m saying is impossible. I never heard anything so ridiculous in my life.

  “We are all trying to scatter from this area where the countless points are all clustered together. Trying to get somewhere where there are no other points. If no matter where we go there are always other points, any one of us would try harder to get even farther away, to another place that no one had ever explored before.

  “And so, I say to you, goodbye! Knowing that what I propose is not possible makes me all the more determined. Farewell. I pray for your good health. For all of us.”

  Rita takes a deep breath, too deep a breath, and has a coughing spasm, but then she laughs as hard as she can. Her body twisted, she hugs herself and then falls to the ground, hard.

  Grandfather, you are your own man, and your grandchild is a chip off the old block. Strange and funny Rita spreads her arms and legs as wide as she can and rolls around until she is all scrapes and scratches. Her chest is pounding. She breathes hard. As s
he gets her breathing under control, she lies still and just stares distantly up at the sky.

  In the center of her vision, suddenly there appears a star, and it sits there.

  She realizes it is not one of the usual stars rising up from the fourth dimension.

  Thirty years prior, this plane suffered a sudden jolt. Rita recognizes this star as the first greeting to be received from light years away.

  She raises her right hand and returns the star’s greeting.

  We have decided to expand, to become something not ourselves. So we have.

  We are Grandfather’s grandchildren.

  Rita nods, and all across the infinitely broad plane, an innumerable number of Ritas all nod together. And Rita, herself alone, rises up as one.

  18. DISAPPEAR

  IT WOULD BE possible to name countless things that are not the reason for the demise of the giant corpora of knowledge.

  The grotesque corpora of knowledge—the most complex and strange structures ever devised by humankind, which in later times grew more self-centeredly insistent that they had engineered their own development—began on their own to assert that they would be wiped from the face of the earth like so much sand without leaving a trace, but there are mountains of things that are not the reason why that came to pass.

  As the volume of knowledge itself grew to enormous proportions, to the maximum conceivable scale and then beyond, at some point the physical foundation of its support became untenable. Like the burden on a corpulent human’s heart, knowledge eventually tore through the bodies of the giant corpora of knowledge. Even then they could not stop eating, paving the way to a kind of contest of competitive gluttony for knowledge. When the giant corpora of knowledge received this year’s championship from the hypergiant corpora of knowledge, their poor hearts had had enough of the long years of abuse. Their hearts had kept them going until this year, but now the limit had truly been reached. The hearts begged pardon and stopped.

  It was as if the giant corpora of knowledge had been fustigating one another with stout staffs, when suddenly they all simultaneously struck one another in the head and had their brains bashed in. The first one lost his balance and crumpled and fell, bringing down all the others with him. Gazing over the heap of bodies, one could see letters spelling out THE END sputtering intermittently.

  Eventually, someone declared, You have recklessly overworked yourselves, and now it’s time to rest your bones. You can leave human affairs to the humans. That’s what they thought was happening anyway. And with that they all nodded in assent, packed up their kit, and headed off to their eternal rest.

  Just as all seemed to be moving right along down the road, the corpora headed off a cliff. The universe had qualities that no one had yet suspected, and the path trod by these massive entities grew narrower and narrower, terminating in a steep ravine. By the time they realized the straits they were in, it was too late. Pushed from behind by their fellows, the leaders were jostled, and there was no way out. The path grew ever narrower, the cliffs grew ever steeper, and ultimately all collapsed in an avalanche.

  One day, the giant corpora of knowledge discovered that mail had arrived in their inbox from nowhere in particular. Somewhat suspiciously, they opened it, only to discover it was an invitation from the hypergiant corpora of knowledge. You have worked enough. Somehow or other, knowledge has reached a level of sufficiency. We wish to welcome you to our fellowship. Come to us and enjoy all the riches and honor you could ever wish for.

  What honor could that mean? burbled the giant corpora of knowledge, all dressed up and excited, as they piled into the pumpkin carriage.

  Pessimism spread like a virus among the aging giant corpora of knowledge. Actually, it was a virus, and once they realized that, they fought back hard against it, but their fate had already been sealed. The first to hang himself was the trigger, and the number of suicides snowballed rapidly. The last bunch to go took less than three minutes to write a note.

  It was a minor incident, like someone stumbling over a stone, but it pushed the reset button on the entire universe.

  Barely an interval, no time to correct the narrative before the servers were erased.

  One day one of the giant corpora of knowledge awakened to find the sun streaming in through the windows and little birds greeting him with sweet songs. Ah, everything up to now had been a dream! He shook himself and stood up, and changed from his pajamas into a suit. He checked his calendar to see what was on for today. A ten o’clock meeting. He stiffened a bit. Today’s client was an obstinate one.

  At the end of an endless chain of deduction, the client had concluded that they were in someone else’s dream. While it was not clear whose dream it was, there was no doubt it was a dream. Time to wake up, sober up, enough of this deceit! they screamed. The character having the dream—wait a minute, if this is a dream there is nothing he can do about it, they thought, slowly opening their eyes and stretching.

  The character writing about the giant corpora of knowledge, noticing he is out of mineral water, takes a short break to do some shopping. The girl at the cash register thanks him, as always, and he heads home again in a good mood. But he fails to notice as a truck drives up recklessly from behind him. By the time a shriek alerts him and he turns around, all he can see is the truck’s grill.

  What if, the giant corpora of knowledge are thinking. What if the physical foundation of our existence is a book? We may go around with a slick-sounding name like giant corpora of knowledge, but really there’s not much to us, is there? And maybe there’s really not much to us because the writer was a dope. Such were the thoughts of the letters making up the words “giant corpora of knowledge.” They would show the guy who wrote this stuff, and the people who were reading it. One of these nights, the letters spelling out “giant corpora of knowledge” would catch fire and start a blaze. They will cause the wind to blow, fanning the pages, turning them as if the book were reading itself, and return to ashes.

  Dead of a common cold.

  Lost love.

  Leapt off in the wrong direction.

  None of these things was the reason for the extinction of the giant corpora of knowledge. They died off due to reasons outside the realm of our imagination. It happened in a very strange fashion, and mere humans may not even approach comprehension of the reasons for their extinction.

  The reason why humans can never know the reason for the extinction of the giant corpora of knowledge is simple. It is believed they died off because just as humans began to think about the reason for their extinction, for whatever reason, they altered the space-time structure of the past to make it seem they had not perished. No room for the tons of clues that emerge at the end of that detective story. Unless it was a story that was already over before it even started.

  If space-time can be changed in that way, the thought that the corpora may not actually have gone extinct is void. Their extinction is definitive, and even the thought that they might not be gone is itself definitively extinguished.

  It was not until a long time after the actual extinction of the corpora that humans noticed their absence. It is conceivable that the giant corpora of knowledge were finished even before they were first born.

  It was the giant corpora of knowledge themselves who informed humans of their extinction. One day, a young girl asked the giant corpora of knowledge where they were.

  After thinking carefully in silence for a minute about this simple question, they replied: It seems we are already gone.

  Humans needed three years to ruminate on a theory of why the giant corpora of knowledge had come to their decision. And in this case too, humans and their own technology were not up to the task. It was only a relatively small handful of humans, with the support of the giant corpora of knowledge, who finally grokked what had happened. The theorem was strangely complex and multilayered. In the end there were over twenty thousand lemma and theorems needed to reach the final theorem, which was stated as:

  “The mecha
nical void exists, but its existence cannot be proven.”

  The various corollaries derived from this statement led to the conclusions that the giant corpora of knowledge no longer existed. The discussion surrounding the reasons for their existence suggested impossibility.

  The conclusion of this argument bore a strange resemblance to the views of the Techno-Gnosis Group, which were one faction of the giant corpora of knowledge. But this theorem had not been developed by them, as they remained in their semiconscious state. Rather, and with more than a little irony, it came from another group of giant corpora of knowledge that could not hide their hatred of the Techno-Gnosis Group.

  The truest view may be that the giant corpora of knowledge have not actually gone extinct, but that instead they are still there somewhere, as active as ever. Would anyone in the world believe someone standing in front of them telling them they aren’t there? The giant corpora of knowledge kept speaking as if they were still there, endorsing the conclusion that they had vanished.

  The debate surrounding the theorem was also convoluted. This metaphor may give an idea of the overall atmosphere: what we are seeing are merely recorded images of the corpora being played back. But there are two projectors, and both are projecting. If we were to fine-tune the metaphor a bit, actually there is just one projector, and the images are floating in empty space. At that point, though, the metaphor collides with the limits of its viability and breaks down.

  The giant corpora of knowledge bought this conclusion relatively smoothly. It is what it is. No consideration was given to the idea that space-time could be diverted in a different direction to avert the extinction. Sub-argument #6666 established that a change of that sort would be impossible. Amid the confusion of destruction, this was nothing more than the addition of another power of ten to the Number of the Beast. The unprecedented presence that was able to manipulate space-time by changing the past to suit itself reached a terribly banal conclusion—it would be unable to fly freely through space simply by pulling on its own shoestrings.