Alan Hom, 1924-1951

The Hom set was named for the Chinese-American mathematician Alan Hom, who worked with MacLane and Eilenberg and did pioneering work on profunctors.  Hom was killed in the Korean war.

Euphraneid

(4/5 stars) The Euphraneid is a collection of Book of Mormon pseudepigrapha. They were ostensibly written by Moroni, the son of Mormon, during and after his work on the Nephite record. The title is Greek: euphrano means “to shout for joy”; it seems to be asserting that the name Moroni derives from the Hebrew ranan רנן, “to shout or rejoice”.

The stories are short; they contradict one another and the Book of Mormon account itself—but taken together, they make a satisfying whole. If you don’t want spoilers, stop reading here; I summarize my favorites below.

Some stories don’t seem to have a point. In one, Moroni has barely survived a plague—perhaps diphtheria—and returns to his home late in the spring. He’s told in a dream that he has to get gold to make plates, but knows he needs to plow and plant his fields. Trusting in a miracle, he goes up into the mountains to gather gold from alluvial streambeds using fleece to trap the flakes. He spends several weeks setting up traps; when he fears he can’t wait any longer to plant, he returns to find his fields plowed and planted, but also finds his thatch hut occupied by a Lamanite scouting party. That night, he creeps up to the hut, shouts a Lamanite insult and stabs a man through one of the gaps in the wall. In the confusion, the Lamanites kill each other. After burying the men, he returns to the mountain and hangs out the gold-laden fleece to dry. It’s never made clear who plowed the field; I like to think it was the Three Nephites.

There’s an amusing chapter in the Book of Mormon where Mormon repeats his own name twelve times in a single chapter, and five times in a single verse, all without mentioning himself once. Nephi, son of Helaman, writes about how his father named him after the titan-hero Nephi that came out of Jerusalem. In the Euphraneid, Moroni keeps up the tradition, playing the role of a bard singing war stories about Captain Moroni and his men.

My favorite was the retelling of the king-men plot. In this version, Amalickiah is almost vampiric: a landed lord of old blood, meting out extraordinarily harsh punishments in his jurisdiction. Both amputation and impaling were not uncommon, though any punishment could be avoided by paying a large enough fine; by paying a regular tribute, a wealthy man could be immune to the law. Amalickiah’s oath to drink Moroni’s blood was not unusual; he’d done the same to many other enemies, though where he did not have power he sought to gain it by intrigue rather than by force. Like Rasputin, he miraculously survived multiple assassination attempts. I thought it particularly fitting that Teancum played Van Helsing to Amalickiah’s Dracula, killing him with a stake through his heart.

The last story details how, after sixteen years of wandering, Moroni returns to Cumorah. He describes the lay of the land he spent so many months scouting, the remnants of the great battle he fought. He sees the bowl-shaped depression where Cumenihah’s battalion allowed themselves to be drawn out by the Lamanites using a feinted retreat. He comes to the stones of Cumorah’s fortifications and sights along them to the hidden mouth of the cave where he has stored the records; when he arrives, he finds the door open, the room empty and dark, the treasures gone. Moroni feels as though he is drowning, unable to breathe. Then a calm comes over him as he sees that this is not his Cumorah, but another; he is on a different continent, among the ruins of a different civilization that collapsed in a different great battle, their historian’s treasures plundered. His book yet lies hidden and safe, somewhere among the hills.

I said it was the last story, but here even the pseudepigrapha have dubious appendices. The first is told in third person, with no indication of who the narrator is. As it begins, Moroni is carrying the plates, struggling to reach the cave. Six men are in pursuit; they slip on scree. Moroni throws aside the vegetation masking the entrance and makes his way down long narrow passage into a room. He strides past implements, vestments, precious metals and stones, drops the plates on a table, grabs Laban’s sword from the wall. He meets the pursuers halfway down the hall; they measure swords, then attack. Moroni smites the first and he falls dead; another advances and contends with him. This one also falls by his sword; a third then steps forth and meets the same fate; a fourth afterwards contends with him, but in the struggle with the fourth, Moroni, being exhausted, is killed. The remaining two tread on Moroni as they enter the cave; they slip on blood bathing the smooth rock. The cave is empty; they curse and rage. When they turn to leave, even Moroni’s sword is gone.

The final appendix is an alternate account of Joseph Smith getting plates from Moroni. It’s all wrong, but very familiar at the same time.

Joseph has recently acquired a seerstone made of polished quartz; to get some time alone, he takes a deer-trail through the woods. He reaches a widening of the trail where the canopy does not block the sunlight and he takes the stone from his pocket; it is late afternoon, and he starts a small fire by focusing the light with the lens. He’s alarmed to find that he’s unable to extinguish it and fears that he’ll start the forest on fire, but it doesn’t spread. He notices that the wood is not consumed, so immediately bends down to remove his shoes.

At this point, he’s seized from behind; despite Joseph’s fame as a grappler, his assailant was better, or at least good enough to keep Joseph from escaping. “Put out the fire!” the man commands. Joseph replies “I have tried; I cannot.” The assailant mutters some words in a language Joseph does not speak but recognizes, and he calls Moroni by name and tells Moroni to release him. Moroni refuses, citing his fear of seers. Joseph tells Moroni that he knows Moroni carries the second sign and that he should hand it over. At that, Moroni releases him.  Joseph, knowing Moroni is still skittish, doesn’t move.

“I have the first sign already,” says Joseph, and tells Moroni to check his pocket. Moroni puts his hand in to touch the stone but quickly pulls his hand back as though burned. Moroni says of the plates, “They are so heavy.”

Only now does Joseph turn and look at Moroni. He’s about 5 feet 8 or 9 inches tall and heavy set; his face is as large and hawk-like. He is dressed in a suit of brown woolen clothes, his hair and beard white, and has on his back a sort of knapsack with something in it, shaped like a book. He is miserable and worn. Joseph summons himself and commands Moroni to give it to him immediately, or “this very moment either you or I shall die.”

At this, Moroni hands over the plates; he is ecstatic to be rid of them. As Joseph receives them, the fire goes out, and then a woman steps out of the darkness and calls Joseph by name.

It is Sallie Chase, sister to Willard, and a seer. She casts a hex on them with her green glass, then takes the plates. Mocking Joseph, she takes the seerstone from his pocket and places it into silver frames next to her green one.

At this, a column of light appears from heaven and a personage appears; walls of fire spring up on either side of the trail and force her off it. She is forced to leave the plates and stones behind. Joseph is scolded, but allowed to keep the items. Moroni fades from view and the personage ascends to heaven.

I didn’t give the book five stars because there were parts that tended to drag on a bit, but there were plenty of fun tales, too. Overall well worth the read.

The Sum of Forking Paths

Paracelsus threw the rose into the fire; it bent on impact, recoiled, and fell deeper among the logs. In the gold-orange light, the stem was already black. The husk began to shrivel and split as the sap boiled away on its surface. The petals blistered and blackened and fell. The five-fingered star beneath them danced subtly, swaying in the brittle heat. For nearly an hour it lay visibly unchanged save for a gradual loss of hue and a universal greyness, then fell into three large pieces as the log crumbled around it. The ashes glowed orange, then gradually dimmed; the last visible flash of light burst outward from the remains of the stem. Like all light, it carried within it a timepiece.

Once, when the clock read noon, it traveled without hesitation in a straight path to my retina. Once it took another course, only to bend around a molecule of nitrogen and reach the same destination.

Once it traced the signature of a man no one remembers.

Once, at half past three, it decayed into a tiny spark and its abominable opposite image; their mutual horrified fascination drew them together, each a moth in the other’s flame. The last visible flash of light from their fiery consummation was indistinguishable from the one that spawned them.

Once, when the clock ticked in the silence just before dawn, the light decayed into two mirrored worlds, somewhat better than ours due to the fact that I was never born there. Both worlds were consumed by mirrored dragons before collapsing back into the chaos from which they arose; all that remained was an orange flash of light.

Once it traveled to a far distant galaxy, reflected off the waters of a billion worlds and witnessed the death of a thousand stars before returning to the small room in which we sat.

Once it transcribed a short story of Borges in his own cramped scrawl, the six versions he discarded, the corrupt Russian translation by Nabokov, and a version in which the second-to-last word was illegible.

Once it traveled every path, each in its time; once it became and destroyed every possible world. All these summed to form what was: I saw an orange flash, and in that moment, I was enlightened.

Formal axiomatic divination

It’s well known that “in Xanadu did Kublai Khan a stately pleasure dome decree,” but his true legacy is the field of formal axiomatic divination. In 1279, Khan sought an auspicious date on which to begin construction of the palace. He consulted each of his twelve astrologers separately and without warning; unsurprisingly, he received twelve different answers. Khan flew into a rage and said that until the astrologer’s craft was as precise as that of his masons and carpenters, they were banished from his presence.

Kublai Khan died in 1294 and his successor Temur Khan was convinced to reinstate the astrologers. Despite this, the young mathematician Zhu Shijie took up the old Khan’s challenge in 1305. Zhu had already completed two enormously influential mathematical texts: Introduction to Mathematical Studies, published in 1299, and True reflections of the four unknowns, published in 1303. This latter work included a table of “the ancient method of powers”, now known as Pascal’s triangle, and Zhu used it extensively in his analysis of polynomials in up to four unknowns.

In turning to the analysis of divination, Zhu naturally focused his attention on the I Ching. The first step in performing an I Ching divination is casting coins or yarrow stalks to construct a series of hexagrams. In 1308, Zhu published his treatise on probability theory, Path of the falling stone. It included an analysis of the probability for generating each hexagram as well as betting strategies for several popular games of chance. Using his techniques, Zhu became quite wealthy and began to travel; it was during this period that he was exposed to the work of the mathematicians in northern China. In the preface to True reflections, Mo Ruo writes that “Zhu Shijie of Yan-shan became famous as a mathematician. He travelled widely for more than twenty years and the number of those who came to be taught by him increased each day.”

Zhu worked for nearly a decade on the subsequent problem, that of interpreting a series of hexagrams. Hexagrams themselves are generated one bit at a time by looking at remainders modulo four of random handfuls of yarrow stalks; the four outcomes either specify the next bit directly or in terms of the previous bit. These latter rules give I Ching its subtitle, The Book of Changes. For mystical reasons, Zhu asserted that the proper interpretation of a series of hexagrams should also be given by a set of changes, but for years he could find no reason to prefer one set of changes to any other. However, in 1316, Zhu wrote to Yang Hui:

“I dreamed that I was summoned to the royal palace. As I stepped upon the threshold, the sun burst forth over the gilded tile; I was blinded and, overcome, I fell to my knees. I lifted my hand to shield my eyes from its brilliance, and the Emperor himself took it and raised me up. To my surprise, he changed his form as I watched; he became so much like me that I thought I was looking in a mirror.

“‘How can this be?’ I cried. He laughed and took the form of a phoenix; I fell back from the flames as he ascended to heaven, then sorrowed as he dove toward the Golden Water River, for the water would surely quench the bird. Yet before reaching the water, he took the form of an eel, dove into the river and swam to the bank; he wriggled ashore, then took the form of a seed, which sank into the earth and grew into a mighty tree. Finally he took his own form again and spoke to me: ‘I rule all things; things above the earth and in the earth and under the earth, land and sea and sky. I can rule all these because I rule myself.’

“I woke and wondered at the singularity of the vision; when my mind reeled in amazement and could stand no more, it retreated to the familiar problem of the tables of changes. It suddenly occurred to me that as the Emperor could take any form, there could be a table of changes that could take the form of any other. Once I had conceived the idea, the implementation was straightforward.”

The rest of the letter has been lost, but Yang Hui described the broad form of the changes in a letter to a friend; the Imperial Changes were a set of changes that we now recognize as a Turing-complete programming language, nearly seven hundred years before Turing. It was a type of register machine similar to Melzak’s model, where seeds were ‘planted’ in pits; the lists of hexagrams generated by the yarrow straws were the programs, and the result of the computation was taken as the interpretation of the casting. Zhu recognized that some programs never stopped–some went into infinite loops, some grew without bound, and some behaved so erratically he couldn’t decide whether they would ever give an interpretation.

Given his fascination with probabilities, it was natural that Zhu would consider the probability that a string of hexagrams had an interpretation. We do not have Zhu’s reasoning, only an excerpt from his conclusion: “The probability that a list of hexagrams has an interpretation is a secret beyond the power of fate to reveal.” It may be that Zhu anticipated Chaitin’s proof of the algorithmic randomness of this probability as well.

All of Zhu’s works were lost soon after they were published; True reflections survived in a corrupted form through Korean (1433 AD) and Japanese (1658 AD) translations and was reintroduced to China only in the nineteenth century. One wonders what the world might have been like had the Imperial Changes been understood and exploited. We suppose it is a secret beyond the power of fate to reveal.

Lazulinos

Lazulinos are quasiparticles in a naturally occurring Bose-Einstein condensate first described in 1977 by the Scottish physicist Alexander Craigie while at the University of Lahore [3]. The quasiparticles are weakly bound by an interaction for which neither the position nor number operator commutes with the Hamiltonian. A measurement of a lazulino’s position will cause the condensate to go into a superposition of number states, and a subsequent measurement of the population will return a random number; also, counting the lazulinos at two different times will likely give different results.

Their name derives from the stone lapis lazuli and means, roughly, “little blue stone”. Lazulinos are so named because even though the crystals in which they arise absorb visible light, and would otherwise be jet black, they lose energy through surface plasmons in the form of near-ultraviolet photons, with visible peaks at 380, 402, and 417nm. Optical interference imparts a “laser speckle” quality to the emitted light; Craigie described the effect in a famously poetic way: “Their colour is the blue that we are permitted to see only in our dreams”. What makes lazulinos particularly interesting is that they are massive and macroscopic. Since the number operator does not commute with the Hamiltonian, lazulinos themselves do not have a well-defined mass; if the population is N, then the mass of any particular lazulino is m/N, where m is the total mass of the condensate.

In a recent follow-up to the “quantum mirage” experiment [2], Don Eigler’s group at IBM used a scanning tunneling microscope to implement “quantum mancala”—picking up the lazulino ‘stones’ in a particular location usually changes the number of stones, so the strategy for winning becomes much more complicated. In order to pick up a fixed number of stones, you must choose a superposition of locations [1].

1. C.P. Lutz and D.M. Eigler, “Quantum Mancala: Manipulating Lazulino Condensates,” Nature 465, 132 (2010).
2. H.C. Manoharan, C.P. Lutz and D.M. Eigler, “Quantum Mirages: The Coherent Projection of Electronic Structure,” Nature 403, 512 (2000). Images available at http://www.almaden.ibm.com/almaden/media/image_mirage.html
3. A. Craigie, “Surface plasmons in cobalt-doped Y₃Al₅O₁₂,” Phys. Rev. D 15 (1977). Also available at http://tinyurl.com/35oyrnd.

Life imitates Art

In “Tlön, Uqbar, Orbis Tertius“, Borges describes a group of dedicated people who describe a world, Tlön, in such detail and produce enough forged artifacts that the whole world adopts their vision and begins to convert itself into Tlön. I attributed a farcical version of Borges’ story “Death and the Compass” to a pseudonymous Umberto Eco in the last post; in that story, the detective’s supposition that there is a pattern induces his prey to begin using the pattern in order to entrap him. In fact, Eco borrowed that idea for his own detective story, The Name of the Rose. The Bible formed the basis for the production of thousands of fraudulent relics, and the Book of Mormon inspired Mark Hoffmann to create forgeries of letters from the early Mormon community (though his were designed to destroy the community instead of build it). Cellphones look like Star Trek communicators, and Bluetooth headsets look like Uhura’s earpiece. Of course, there’s all the merchandising from Star Wars and Lord of the Rings. And my brother David just gave my brother Doug a set of the six signs from Susan Cooper’s The Dark is Rising sequence.

What story has motivated you to create your own artifacts?

Creative ways to ask a girl on a date

In Utah, it is expected that high-school students and undergraduates will come up with extravagant ways, often including horrible puns, to ask a girl out on a date and to reply to such an invitation. My cousin’s family, having just moved there, asked on the family list for ideas. My response:

My date read the “police beat” religiously, so I got Umberto Eco to write a story about a detective on the BYU campus police force who believes that two crimes (involving the vandalism of FARMS researchers’ webpages with screeds linking kabbalah to quantum field theory) are related even though, in fact, they are not. The detective was responsible for a student’s conviction of violating the Honor Code and eventual expulsion. The student learns of the detective’s interest and begins to commit crimes that fit the imagined pattern; the detective predicts where the final crime will be committed and stakes out the place, but because the student was expecting him, he is captured by the student and his friends and forced to wear U of U paraphernalia.

I had it published pseudonymously in the Daily Herald, and then used my network of friends to commit 137 minor crimes across campus for the next six months prior to the dance that spelled out “[Name], will you go to the dance with me?” I waited at the place where the point of the question mark would be, but she never showed up.

She did, however, wire my brakes to my horn and leave a sheet with a big “YES” painted on it under the hood. This would have been amusing had the new wiring not caused electrical arcing. The sheet caught fire, which caught other things on fire, which eventually burned out the interior of the car. It wasn’t too much of a loss, though, because it was a 1973 Honda Civic that I’d bought from a graduating senior for $200.

Anyway, neither of us wanted to dance all that much in the first place, so we went down to the underground laser lab where my roommate worked and then explored the steam tunnels.

The Murder of Asher Ben-Judah

Here’s a story I wrote; it’s inspired by Borges’ collection of stories “A Universal History of Infamy.”

The Murder of Asher Ben-Judah

In the fourth year of the reign of Nebuchadnezzar II, Egypt successfully repelled the invasion by Babylon. Believing Babylon to be weakened, Jehoiakim of Jerusalem stopped paying tribute to Babylon, took a pro-Egyptian position, and promptly died. His son Jeconiah chose to continue the policy; one hundred days later, he was deposed by Nebuchadnezzar II for rebellion. The Babylonian king sacked the temple, took captive all the nobility and craftsmen who had not fled the city—some ten thousand people—and carried them off to Babylon; the prophet Ezekiel was among them. Before leaving, perhaps mockingly, Nebuchadnezzar annointed Jeconiah’s uncle Mattaniah, clothed him in the robes of kingship, and gave him the new name “Righteousness of the LORD.”

Despite the destruction, the harvest that year was a good one for farmers, and the sale of the excess bought capital for rebuilding the city. Those wise and wealthy enough to have fled Jerusalem with their property in anticipation of the inevitable response to Jehoiakim’s stupidity returned; among them was the ward boss Asher Ben-Judah. Asher was a master at organizing labor; he was often and fruitfully compared to Father Jacob’s father-in-law for having the cunning to convince a man to work fourteen years in the hope of being paid someday. However, when cunning failed, Asher was not above resorting to other motivators: he was also a master at organizing crime. If one were to speak to a particular man in the bazaar, he would recite a list of Asher’s prices:

• Punching – 2 shekels,
• Both Eyes Blackened – 4 shekels,
• Nose & Jaw Broken – 10 shekels,
• Ear Chawed Off – 15 shekels,
• Leg Or Arm Broken – 19 shekels,
• Stab – 25 shekels,
• Doing the Job – 100 shekels and up.

As the armies of Babylon flooded the country, Asher came to rest in the mountains of Ararat. A generation before, the Arartian king Rusa II had built more cities than Solomon, Ramses, Semiramis and Sargon put together; the blind arches of Rusahinili and Teishebaini rivaled the fortifications of Ninevah. Asher knew there would be plenty of work for masons in rebuilding Jerusalem after Babylon was through with it.

Another household returning to Jerusalem that year was that of Asher’s second cousin “Jawbone” Ben-Samson, a merchant dealing in precious metals and a smith in his own right, having received the secrets of metallurgy from his fathers. Ben-Samson had chosen to find refuge in Egypt, where Babylon could not follow, and returned with artifacts of gold, silver, brass, and steel.

Though Asher cared nothing for working metal, he was the firstborn and had inherited the sword forged by their great-grandfather; the iron was cast down from heaven and laid waste to a forest near Damascus. Such iron was very rare and very valuable, since it was pure enough to be strengthened by forging in charcoal; iron extracted from ore already had too much of the black ash in it, and would become brittle.

It’s unclear what happened to spark Ben-Samson’s madness. He began to accuse the king of plotting against Babylon; the king, who owed his throne to Nebuchadnezzar’s grace, ordered his death, but Ben-Samson escaped to the desert. He began to forget key metallurgical processes; he sent his sons to Asher to coerce him into giving them their great-grandfather’s records. Asher turned them away, but they returned and attempted to buy the records; insulted at the prospect of selling his birthright, Asher told his men to kill the intruders. “Jawbone” Ben-Samson was not so named because he was a weakling, and his sons lived up to their name: they fought off the thugs and escaped, but in the scuffle they dropped the keys to their family’s treasury. Since neither Ben-Samson nor his sons could reenter the city to claim their property, Asher became the second-richest man in Jerusalem.

Asher, dressed in his finest, went out on the town to celebrate. He bought everyone drinks at the ward tavern and used his favorite prostitute; near the end of the third watch he stumbled out the door towards home. Asher Ben-Judah was found stripped and decapitated the next day; his sword and his great grandfather’s records were missing. Neither Ben-Samson nor his sons ever returned to Jerusalem.

Aleph and Omega

I shut my eyes — I opened them. Then I saw the Aleph.

I arrive now at the ineffable core of my story. And here begins my despair as a writer. All language is a set of symbols whose use among its speakers assumes a shared past. How, then, can I translate into words the limitless Aleph, which my floundering mind can scarcely encompass? Mystics, faced with the same problem, fall back on symbols: to signify the godhead, one Persian speaks of a bird that somehow is all birds; Alanus de Insulis, of a sphere whose center is everywhere and circumference is nowhere; Ezekiel, of a four-faced angel who at one and the same time moves east and west, north and south. (Not in vain do I recall these inconceivable analogies; they bear some relation to the Aleph.) Perhaps the gods might grant me a similar metaphor, but then this account would become contaminated by literature, by fiction. Really, what I want to do is impossible, for any listing of an endless series is doomed to be infinitesimal. In that single gigantic instant I saw millions of acts both delightful and awful; not one of them occupied the same point in space, without overlapping or transparency. What my eyes beheld was simultaneous, but what I shall now write down will be successive, because language is successive. Nonetheless, I’ll try to recollect what I can.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny — Philemon Holland’s — and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.

I felt infinite wonder, infinite pity…

(Jorge Luis Borges, The Aleph)

A finitely-refutable question is one of the form, “Does property X holds for all natural numbers?” Any mathematical question admitting a proof or disproof is in this category. If you believe the ideas of digital physics, then any question about the behavior of some portion of the universe is in this category. We can encode any finitely refutable question as a program that iterates through the natural numbers and checks to see if it’s a counterexample. If so, it halts; if not, it goes to the next number.

The halting probability of a universal Turing machine is a number between zero and one. Given the first bits of this number, there is a program that will compute which -bit programs halt and which don’t. Assuming digital physics, all those things Borges wrote about in the Aleph are in the Omega. There’s a trivial way–the Omega is a normal number, so every sequence of digits appears infinitely often–but there’s a more refined way: ask any finitely-refutable question using an -bit program and the first bits of Omega contain the proper information to compute the answer.

The bits of Omega are pure information; they can’t be computed from a fixed-size program, like the bits of can.