Self-Reference Engine

by Toh EnJoe

  Press coverage of this strange notion was poor, perhaps because the statements were less than completely understood. In a nutshell, Professor Moriarty’s crimes were a figure of speech; no one desired an explanation that required Professor Moriarty to traverse space-time. This was just a peculiar coincidence. To embellish would be inelegant.

  The SF fans, seen as having the disadvantage in this situation, tried to shift their position, but the mystery faction wasted no time in implacably trumpeting the facts of the incident.

  For whatever reason, the universe in which we now live has a structure bearing a strong resemblance to the universe Conan Doyle created. Professor Moriarty may be nothing more than a creation of Conan Doyle’s, but our universe is one in which a theorem like the one he demonstrated might exist. This suggests strongly that we ourselves were in fact written by someone. This quality is well known among SF fans as a “written space,” they went on, but by about that time no one was still listening.

  This refrain contributed valuable corroboration to the observations concerning why the SF fans were being driven to the brink of extinction, but few were deeply impressed by this interpretation.

  And the mathematicians responded sincerely, as mathematicians should, displaying their mathematicianness for all to see: if, for argument’s sake, this were a different universe, mathematical truths should still be strict truths—for the introduction of a nonsensical new universe that simply had more theorems, no approval could be given.

  Even so, it was hard to believe that such a concise and lucid theorem could have gone unknown until now. We have certainly been tricked by something, the SF fans responded.

  Mathematical truths cannot be misrepresented, the mathematicians said, unable to contain their annoyance. But a theorem might be able to camouflage itself as truth by causing truth judgment neurons to fire, the SF fans asserted, and the mathematicians categorized them as the sort of opponents one needn’t take seriously.

  This sort of sterile argument failed to hold people’s interest for long, and soon a feeling set in that something was not right. The things the SF fans said were certainly ridiculous, but still there was the widespread feeling that someone was trying to put something over on somebody, and they too started to be aware of it.

  The theorem itself was fine. It was practically self-evident. But what about the idea that twenty-six mathematicians had all thought of it at the same time, written it up at the same time, submitted their manuscripts at the same time? Had someone been standing on the sidelines with a stopwatch, checking their times?

  That is no more than a coincidental prank, nothing that science needs to meddle with, the mathematicians insisted curtly. Extremely improbable, perhaps, but the probability of occurrence is not zero. And if it’s not zero, that means it can happen. They themselves dealt with phenomena whose probability was actually zero. This was nothing compared with that. And for the third time it was repeated that the twenty-six mathematicians were not trying to put one over on the world by publishing as new a theorem they had all known about for a long time already.

  So, then, what did it mean?

  No one could answer that question. It had simply happened.

  And three weeks after the theorem was published, the world was attacked by the Event.

  Even now it is not clear exactly what happened at that moment.

  A night passed, then a morning came. One night, all of a sudden, the theorem simply shattered into so many meaningless strings of characters. It was as if the fluctuations of numberless particles formed themselves by chance into letters and were scattered in the air.

  It is not even clear whether the history I recorded as belonging to this episode has any continuity with the history we now know.

  The present time matrix can be traced back to an inversion of space-time that occurred 10-20 seconds after the Event. Physicists now predict that sometime in the next ten years, research will allow us to understand the form of the universe 10-24 seconds after the Event. For now, though, the route to the instant of the Event itself is closed, beyond hope.

  There are many theories about what exactly happened in the instant of the Event.

  One idea is that in that instant our universe was shattered into innumerable shards of universes, which blew away in random directions.

  Another idea is that an extradimensional universe collided with our universe. Another idea is that our universe was shredded into countless shards as it bubbled up from the vacuum. Yet another idea is that our universe itself was a bubble born as a structure camouflaged from the very beginning, a repeated oscillation of creation and annihilation.

  Of these ideas, one includes the prediction that at approximately 289 seconds after the Event, we will enter a space-time realm where the A to Z Theorem will once again be valid.

  At this point, we have no basis on which to compare and debate the strengths or weaknesses of any of these theories. Each idea has its share of the sort of elegance theoreticians aspire to. Just which of these beauties is in agreement with the beauty of our present space-time, which is nearing a peak of disorder, remains completely beyond our grasp.

  I like this fable:

  There once was a book in which the countless universes were recorded. A librarian spilled coffee on the book, stood up abruptly, and dropped it. The book, which was very old, split apart on impact, and countless pages wafted up into the air. The clueless librarian anxiously attempted to collect the pages and put them back, but had no idea in what order to put them.

  Now, fables do not ordinarily leave the realm of fabulation, but the nice thing about this fable is that it is said that the librarian had the book open to the pages on which were recorded the canonical works of Sherlock Holmes. The page on which the librarian spilled the coffee was “The Final Problem,” erasing the record of Moriarty’s fall from Reichenbach Falls so it never happened. With that abrupt change, Moriarty was suddenly enlightened. He realized that he was in fact a character written in a book, and he resolved to devote himself to communicating to us that he had difficulty permitting himself to engage in the kinds of criminal behavior ascribed to him as the Napoleon of Crime.

  But of course, a fable is only a fable.

  For myself, I like to imagine that the librarian is, even now, desperate to restore the book to its original order. It may seem difficult to reorder infinite pages, but I think it is a more constructive approach than the next one.

  I mean, more than imagining a scene where the book simply fell, on its own, with nobody there in the library, and it scattered about crazily in countless bits, and it laughed.

  It would not be wrong here to note that, since that time, a certain phenomenon has occurred from time to time that perhaps ought to be called the obverse of a similar truth. About two centuries ago, a group of twenty-five physicists garnered attention when they published the B to Z Theorem, which was known at the time as the world’s ultimate theorem. It is all but forgotten now, but it followed the same path as the A to Z Theorem. For one thing, it is not well known, but there was a public that could follow the ins and outs of that kind of theorem. Another reason is that it was followed soon after by the C to Z Theorem. Then, once the D to Z Theorem emerged, its shadow was even paler, and with the E to Z Theorem, one hesitates to wager whether the discussion is even worth pursuing. Of course, one is free to assert this is merely the progress of theory: the appearance and annihilation of strange truths, advanced by a series of agreements known to be destined to turn to dust; this becomes the problem of questioning the truth of the concept of truth.

  Even so, there is a reason why, recently, media interest in the ultimate theorem has revived. The theory currently considered the latest and most consequential is actually the T to Z Theorem. The observations just described regarding the shape of space-time following the instant of the Event are derived from this theorem. If this alphabetic progression of theorems continues like this, renewed by root and branch, before long we will reach the X to Z Theorem, fo
llowed by the Y to Z Theorem. The ultimate member in this progression would be the Z to Z Theorem, or simply the Z Theorem. I like to think this will simply represent the theory of ultimate truth with no particular basis whatsoever.

  This is a hopeful interpretation of the phenomenon wherein a global truth appears suddenly, correctly, self-evidently, and simultaneously in the minds of multiple people, and the reason why the initials of the last names of the authors would contract in order, from A to Z. While we continue to be made fools of by someone or something, we continue to believe we are progressing, if only haltingly, in the direction of the ultimate theorem, and somehow this comforts us. At least I think that is the most convincing explanation of this strange phenomenon.

  But of course, there is an obvious problem with the idea that the Z Theorem will be the ultimate theorem. If the Z Theorem is the true ultimate theorem, which Z Theorem, produced by which person whose last name begins with Z, will be the ultimate theorem? The A to Z Theorem won attention because it was discovered simultaneously by twenty-six mathematicians. The same was true of the theorems that followed. Of course, there was also the clear marker that their results were so simple. How sure can we be, though, that the Z Theorem we now expect to appear will also be simple? Theory or theorem, at some level all must be simple and clear and just as they are.

  I would love to encounter such a theorem. And I hope it would betray my expectations, render the current discussion meaningless, and be overwhelmed by loud laughter. But this hope of mine is being supplanted by an anxiety that we may never reach that point.

  A landscape in which texts containing truths are swallowed up in a sea of papers. I am imagining, for example, a single strange molecule that may exist in the midst of such a sea.

  Or else, it could be that when the Z to Z Theorem ultimately appears, and truth is once again upended, this disturbance will simply blow over. It’s fun to think that after that, without theorems or anything like them, the null set may appear, or a Null Set ø Theorem based on that, and from this Null Set ø Theorem the Von Neumann Ordinals: the {ø} Theorem, the {ø ,{ø}} Theorem, the {ø ,{ø, {ø}}} Theorem.

  Given a choice, I would choose to be involved with this last. The ø Theorem points toward the Transfinite Number ω Theorem, which could lead to the ω + 1 Theorem, the ω + 2 Theorem, 2ω Theorem, ωω Theorem, etc., etc., a progression of large cardinal numbers.

  It is just possible that, via this method, we will reach the realm of theories incomprehensible except with inordinately massive intelligence.

  And then one day, at the pinnacle of the limit of this progression, a grave voice will intone that the truth is “42” or some such. Or we will hear the echoes of Professor Moriarty laughing that truth is the Binomial Theorem. And then, in that instant, Sherlock Holmes will interrupt that laughter, and he and the professor will plunge down the waterfall.

  Without end.

  And perhaps forever. Ad infinitum.

  04. GROUND 256

  THE BOOKCASE STANDS on my body.

  With both hands I try to lift it, but I can barely budge it. My strength is obviously no match for a bookcase, but with it propped up on my arm I am able to twist my body and roll out from between the futon and the bookcase.

  Pressing down my shoulder and turning my left hand, I look up at the ceiling. The bookcase is still growing from the ceiling, so of course there is no way I, my muscles still drowsy, would be able to move it. This enormous, ornately carved piece of furniture is sprouting from the ceiling, and it is more than a little bit frightening to think that its entire weight is resting on me on the bed. At times like this, I usually make some effort to get really scared, but this time I am unable to summon any profound emotion.

  Not because I am not pragmatic. Just a matter of habit.

  For one thing, the bookcase has not yet fully emerged, and for another, it is empty. While this will not be my favorite awakening of my life, it will remain in the category of “not such a bad morning.”

  So, the curtain has risen on today’s menu du jour, and my journey to the kitchen can begin. It has been some time since the door to my room was removed, but a new door stubbornly persists in growing back where the old one once was. Somehow this simply seems to be the nature of things. If I don’t smash it to pieces soon, the door could soon threaten to shut me up in my room.

  Standing beside the bed and stretching, casually hefting something like a crowbar, I begin this morning’s journey to the kitchen.

  As you can see, all kinds of things are growing throughout the house. That said, it still maintains the form of a house. My father built this house originally, by himself. My memories of this house are fond enough, but then unfamiliar houses began invading, haphazardly, almost as if they were ignoring the space completely and attempting to found an entire neighborhood in a single spot. The scene will be easier to picture if you can imagine that.

  The house that is trying to newly emerge seems to have its own rationale, but we keep smashing the newly grown bits, and that seems to be disrupting its plan. Messing with the code while a program is running will cause problems, without question. But we have made up our minds to protect this house, and to protect this village.

  I hack a path toward the kitchen, smashing a chair growing in the hall, then thrashing hangers and desks along the way. Mother is up and getting the day going, brandishing her beloved chainsaw. By the end of the day, the house will finally once again be just one house, but that will be as fleeting as a night’s dream. By the next morning, it will be back like a horribly real nightmare. Somehow the thought of my mother’s life—her daily destruction of the house in order to preserve the house—is very moving. But when I was little I wished her life was a bit more ordinary.

  By the time I reach the kitchen, having dispatched numerous opponents, there are two trickles of blood on my forehead. I failed to notice a pane of glass spanning the hallway and ran straight into it. The thing itself was tangible, but its invisibility made the hallway seem passable.

  A new kitchen table is growing atop my kitchen table, to the point where it is hard to tell which is the original table. Mother also appears at a loss, but she seems to have decided that the first table, which is about the right height to set a plate of fried eggs on, is the original. In this way, a lot of our furniture has actually been swapped out without a second thought, in the way that the molecules of our bodies were swapped out without changing our immutable selves.

  Mother, gripping the frying pan, chainsaw by her side, looks at me and my crowbar thing with a critical gaze.

  “Yuta, I would prefer you didn’t bring dangerous articles like that to the breakfast table.”

  I glance at the chainsaw, but I realize Mother regards it as one of the seven appliances no housewife should be without—like a can opener. I have no strong feelings one way or the other, so I toss the crowbar thing in the direction of the hallway. The time is long since past when people would conceal their crowbarlike tools under the table as they got to know one another.

  I ask about Father, and I am told he has already gone out to the village council. “Major mopping-up operation” is a silly phrase I am already sick of hearing, but now that I am bigger it makes my heart beat stronger in my chest. At some point the grown-ups will certainly do something about this village. That’s what I thought when I was small, and my little heart raced rapidly. But someday turned out to be Well maybe someday, and by now I know that even Christmas comes every year. At Christmastime. But when? Which? Christmas is already over, Grandpa.

  I gulp down the toast and eggs Mother made for me, poke an orange on the floor with my fingertip to make sure it won’t turn into a hermit crab or something, then pick it up. Did this orange really grow on a tree? Or was it really an orange from some other house, one that sprouted here in the night? Or could it be an orange from a tree that had suddenly sprung up here? I bite into it, not thinking too hard about it. Suspicion breeds suspicion, and that is just how it is.

  At som
e point, there is no doubt in my mind, the time will come when I will be confused whether the mother I see before me is the mother who gave birth to me—my own meddlesome mother—or some other mother who came in the night from the other side and grew here.

  When my problems get that big, I will leave them up to the village council, the highest decision-making body in the area, whose to-do list is already growing bigger and bigger.

  I quickly wash my dishes in the sink, stack them neatly, and tell my mother I am going out. I grab a crowbar that is poking out from the wall. I no longer even wonder why these crowbarlike tools seem to be growing everywhere.

  And that is how, once again, on that day, I go out to wreak havoc in the village.

  Armed with the usual crowbars, the youth of the village stagger around in loose teams, wantonly destroying anything they do not remember seeing there before.

  Every morning we head straight for the house of Ms. Tome, who lives away from the village, to save this one-time beauty who is now over eighty. Ms. Tome’s house is a good distance from the village, and every morning we find it in quite a state.

  Ms. Tome lives in an exquisitely constricted state, amid dozens of houses piled up in a jumble of layers, but she herself never seems to mind. She is skilled at folding up her already compact, shrunken frame and waiting quietly for our daily morning rescue mission. We are always careful to extract her from the proliferating furniture without injury, recovering both her and her house.

  The rescued Ms. Tome always releases a weird sound as she stretches herself out straight again, and from some pocket she produces chewy candies to distribute to those of us who have participated in her rescue, one apiece. Then she bows politely, her cheeks peach-pink, to Gen, apparently a one-time suitor, who comes to visit each morning with his head wrapped in a hachimaki bandana.