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Translated from the Japanese by Terry Gallagher.

The Aharonov-Bohm-Curry-Davidson-Eigen-Feigenbaum-Germann-Hamilton-Israel-Jacobson-Kauffman-Lindenbaum-Milnor-Novak-Oppenheimer-Packard-Q-Riemann-Stokes-Tirelson-Ulam-Varadhan-Watts-Xavier-Y.S.-Zurek Theorem—called the A to Z Theorem for short—was, for a brief period about three centuries ago, in some sense the most important theorem in the world.

In some sense. Or possibly in all senses.

Nowadays, this amazing theorem is held to be incorrect, in terms of even elementary mathematics. Hardly anybody ever even thinks about it anymore, because it's just plain wrong.

At a certain instant, on a certain day, in a certain month, in a certain year, twenty-six mathematicians simultaneously thought of this simple but beautiful theorem, affirmed it would be the ultimate theorem that would make their names immortal, wrote papers to the best of their abilities, and all submitted their papers to the same academic journal at roughly the same time.

The separate submissions from writers from A to Z arrived over the course of a few days, and the editor, looking at these virtually identical manuscripts, first checked his calendar. Even allowing for a full measure of variability and a wide deductive scope, there was no way they could all have been written on April 1. And so the editor was left perplexed as to what sort of day he might be experiencing.

Had twenty-six of the world's top mathematicians suddenly formed a conspiracy that each was now seeking to lead? Or was some strange person, with an excess of time and money, playing some prank involving these twenty-six? At any rate, the editor was sure somebody was trying to put one over on him.

Still unsure what kind of joke this would turn out to be, the editor thought first of the reputation of the journal that employed him. The editor was well aware how much mathematicians enjoyed a good joke, and he could only think something out of the ordinary was going on here. Some members of the group that had sent in these manuscripts were themselves members of the publication's editorial board.

The editor got a bit annoyed that they seemed to have extra time on their hands. If they had time enough to be playing jokes like this, they should have time to be planning special issues or doing something about the backlog of articles that needed peer-review. Why instead were they spending time on lousy pranks like this?

There could be some horrible puns buried in the papers or some code that could only be solved by having all twenty-six manuscripts together. He still hadn't thought this through, but he would make them pay for this. Muttering under his breath, still with some sense of expectation, the editor slit open the envelopes and arranged them in order, in nine folders, and began to examine the contents.

Of course the titles were different, and looking at them all just made the editor more irritated. Unbelievably, each title contained the phrase "Binomial Theorem." Who would be writing about the Binomial Theorem in this day and age? Ridiculous! This one was especially laughable: "A Simple Theorem Regarding the Binomial Theorem."

So obvious. The next one was even more ludicrous: "A Remarkable Quality of the Binomial Theorem." I mean, if you're going to try to put something over on someone, couldn't you at least put a little more effort into the title? This stupid title might get past some amateur, but how could anyone think it would impress a seasoned veteran? What did these writers expect from a theorem that had been around since Pascal? Of course, not even the editor thought the Binomial Theorem had had all the juice completely wrung out of it. He was thoroughly convinced of its importance as a tool. But he found it hard to believe it still held the power to engage twenty-six mathematicians, and all at the same time at that.

But somewhere in the corner of his mind, the editor thought faintly, didn't even the greatest principles take the form of the extremely obvious, hidden in plain sight in our quotidian environment, right before our eyes all the time? Like secret messages inscribed on the backs of eyelids. But no matter how you sliced it, there was no such thing as the Binomial Theorem. Just shake your head and shake your way out of that blind alley.

Picking up one of the manuscripts at random, the editor began to read in earnest. Well, as earnest as one can be about papers that were each only about four pages long. It was not long before the editor raised his head again.

He sat in sullen silence, a look of boundless grumpiness on his face, and tossed the paper to the far side of his desk. He grasped his head in both hands and scratched furiously.

What the hell? the editor wondered, staring blankly up at the ceiling. What the hell?

Why had he himself never before thought of this simple but elegant theorem? No more than a few elementary alterations to a four-line formula, but what this theorem expressed was enough to raise goose bumps. But why? Why had no one ever thought of this before? Once this theorem was known, everything, nearly all fields of mathematics, would be supremely clearly, supremely pellucidly, supremely self-evidently transparent.

The editor kicked back his chair and rose, gathering up the papers, and began stomping his feet as if about to run off somewhere. Then he remembered that running off was not what he was supposed to be doing right now, and he plopped back down in his chair.


The above description is not a faithful depiction of historic events, but without question what actually transpired with the editor was something like this. Of course, even I know that what I had to do was to gather as much documentation as possible, meet with as many knowledgeable persons as possible, and get to the bottom of this.

These days, all of the experts of that time are dead, and most of the materials that might illuminate the situation have been lost. Except in unusual circumstances, mathematicians are generally very open creatures, even if they can be a bit eccentric. This theorem, however, was unusual enough to be impossibly unusual. Every person with direct knowledge of the matter zipped their lips. All that remained for certain was a small "errata" that appeared in the journal two months after the publication of the special edition on the Binomial Theorem that included the papers by the twenty-six authors.

At the time, the sole thought in the minds of everyone who had anything to do with this matter was that they had been made fools of. And not by another person.

The simplest way to put this would be: God had made fools of them.


For a theorem to be published and received with enthusiasm only to be found erroneous is not that unusual. If the paper is only four pages long, though, that is another matter. We are not talking about some paper hastily dashed off half jokingly by a crazy graduate student. In this case, we are talking about papers published in a journal, written by people regarded as the top mathematicians of their time, who made submissions at the risk of their own unsullied reputations, and which had passed through the gauntlet of review by other top mathematicians.

To understand this theorem did not require one to be a top-level mathematician or even have a grounding in mathematics. A middle school student could grasp it. Although perhaps it was only mathematicians who imagined the theorem to be a dazzling force that would sweep across all fields of mathematics.

The unbridled enthusiasm that these papers provoked was at fever pitch for about a week. Newspapers, magazines, TV, and Internet were all trumpeting the discovery: the A to Z Theorem was the ultimate theorem, both simple and final, that explained everything there was to know about the world.

The week after that, though, this topic was already no longer such a big deal. Everyone still recognized how fantastic it was, but regrettably it was too simple, too concise. Even primary school students could understand it if you drilled them on it persistently enough. An ultimate truth that anyone can understand at a glance soon becomes something people stop paying much attention to, and everybody starts minding their own business once again.

One esteemed scholar said the theorem would change all of mathematics. But would that make cars run faster or fill your belly? Apparently not. The theorem was incredibly useful in giving us a frightfully transparent view of mathematics. But it was difficult for anyone not a mathematician to grasp just what a transparent view of mathematics could do for you.

Of course, the mathematicians remained enthusiastic, continuing to appear in newspapers and on TV screens feverishly trying to explain this or that, but the specialized vocabulary that came so naturally to them was difficult for the laity to comprehend. How was this different from people thinking they could live their ordinary lives without being able to solve quadratic equations? People were becoming rapidly less aware of the reasoning. According to the mathematicians, this was now more fantastically transparent than ever before. Think of it as like the air we breathe, and the public accepted this and understood it that way.

Popular interest grew explosively, and then in response to the detection of a sudden change in cloud movements, the tone of media reports suddenly changed, as around the time the theorem was announced the media began reporting about a certain organization that was repeating a certain warning.

The group, which was popularly known as Mystery Mania, claimed the theorem was somebody's idea of a bad joke and a crime of hitherto unknown proportions.

The vanguard heralding the warning was a subgroup that held certain works of Arthur Conan Doyle to be sacred canon. They claimed they could finger the criminal in this particular case, and that no process of deduction was even needed. For this group the truth was so obvious it was not even a riddle; they declared they were even embarrassed to be making a statement about it. Broadcasters, who had engaged in the overheating media battle and were flummoxed, even on-air, by what was in fact the oversimplicity of the theorem itself, thought they had nothing to lose by setting up a news conference for the group.

The pompous man who stood up at the news conference as leader of the group seemed uncomfortable with his own height and thin wrists as he rose to the podium, flanked by drab staff members. He set his deerstalker hat and pipe on the lectern and turned his sharp gaze and peculiar nose toward his listeners. At first he stared out at all corners of the audience, but then he averted his eyes meekly. His clothing—things that people don't ordinarily wear and that hung on him like borrowed items—made the man himself seem borrowed. The impact he should have had was completely lost, and the man himself seemed bewildered.

"As I believe you have all already noticed . . ." he began, briefly, lifting his face haughtily, one shoulder raised. He seemed surprised, deep in his heart, by the expressions of irritation at his excessive theatrics and pomposity written in the faces facing him. He lost his composure, and his right hand rose in a gesture of boredom. His speech lost its note of theatricality, and his voice dropped to its natural tone.

"Do you mean to tell me you really didn't notice?" Grasping the lectern with both hands, the man again gazed out over the audience, recognizing the venom in their eyes, and dropped his shoulders.

"Can things really have gone that far?"

As the man's shoulders drooped ever deeper, the crowd began to heckle him: "Just spit it out!" The man straightened up and stared, a look of disbelief on his face.

"Clearly the villain is Professor Moriarty. Really, did no one among you realize this? At the age of twenty-one he published a paper about the Binomial Theorem that confounded the world of mathematics; it was that success that propelled him to his professorship, even though in Victorian London, the Binomial Theorem was just another theorem. But then . . ."

The man cleared his throat loudly.

"Sherlock Holmes tore the bottom out of his thesis. After the professor's famed book The Dynamics of an Asteroid appeared, Holmes was recognized as a genius, and then the two were locked in furious battle. In fact though, it should be difficult to astonish the world of mathematics. For a long time now we have been puzzled by just what it was in those two monographs that led to Moriarty's professorship and just what it was that Holmes found in them. This discussion has been ongoing for decades.

"But now we know. This recently published paper is the paper that Professor Moriarty wrote so long ago, and the current situation is what Holmes revealed and shuddered about!"

The audience, at a loss for how to react to this, whether to laugh or show admiration, were all abuzz, which only made the man suffer more.

"I cannot believe that men of such importance can have been forgotten. We are talking about Holmes, the Sherlock Holmes! The man who, with all his powers, pursued the man known as the Napoleon of Crime, never quite able to catch him, who ultimately resorted to the martial art Bartitsu to physically take down this mystery man. Really, does no one here know him?"

As the man looked out across the audience, the whisperings and murmurings—Does he mean that Holmes?—seemed enough to knock him off his feet. Does he mean Holmes? That one, the one who fought with the dogs? I read those stories when I was a kid. Didn't he die? Yeah, he did. But didn't he come back again? It's just fiction. Is this related to that?

The man observed the clamor and then stepped down from the podium, a little wobbly but himself again. How could it be that these sacred texts had fallen into such neglect? He shook his shoulders and headed toward the exit. The audience, able to sympathize with neither the man's sudden passion nor his equally abrupt dejection, simply watched him walk away.

Summoning the last of his strength, the man stopped in front of the exit and turned back to face the room.

"Clearly, this is Moriarty's crime. That is all we have to say about this." And without a further sound, he slipped through the door and closed it behind him.

For a moment or two of empty time, the sky opened and a beautiful yet terrible light poured down from the heavens, and then the audience came back to their senses. For lack of something better to do, they got up and looked at each other.


The news conference had been like a kyogen comedy of Holmes believers: unbelievably stupid, but it had stimulated interest among the idly curious. Headlines such as COMPLETE CRIME OF PROF. MORIARTY and MORIARTY'S COUNTERSTRIKE ran in various media, and apparently 120 volumes of Moriarty-related detective fiction were published that year.

Without question, the curtain rings down on Professor Moriarty's life in "The Final Problem" at Reichenbach Falls, where both he and Holmes plummet to their deaths. Or at least Holmes seems to fall, but somehow he manages to escape, climb nonchalantly back up the falls, and turn into a new character called Siegelson, returning home via Tibet. At least that was the conventional wisdom among Sherlock Holmes devotees. And if that was the case, what took off from that bit of "wisdom" were the science fiction fans who even at that time were designated an endangered species.

If Holmes were able to fall that way from the waterfall basin and make his way home via Tibet, why should it then be so strange to imagine that his worthy adversary Professor Moriarty was able to fall from the waterfall basin to the present via space-time?

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, followed 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.




Toh EnJoe was born in Hokkaido in 1972. After completing a PhD at the University of Tokyo, he became a researcher in theoretical physics. In 2007 he won the Bungakukai Shinjinsho (Literary World Newcomer's) Prize with "Of the Baseball." That same year brought the publication of his book Self-Reference ENGINE, which caused a sensation in SF circles and which was ranked no. 2 on SF Magazine's list of the best science fiction of the year. Since then, EnJoe has been one of those rare writers comfortable working in both "pure literature" and science fiction. In 2010 his novel U Yu Shi Tan won the Noma Prize for new authors. In 2011 his "This Is a Pen" was nominated for the Akutagawa Prize, and he won Waseda University's Tsubouchi Shouyou Prize. In January 2012, he won the Akutagawa Prize with "Doukeshi no Cyo" (Butterflies of a Harlequin). His other works include Boy's Surface and About Goto.
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