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.
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.
Here’s a design for a passive polarization clock.
The sky is polarized in concentric circles around the sun. The polarization of the southern sky moves through around 180 degrees during daylight hours. It is polarized vertically in the morning, horizontally at noon, and vertically again in the evening.
Align slices of polarized film such that they are parallel to the contours. Any given ray from the center of the sundial outward always hits the contours at the same angle; the angle changes by 360 degrees as the ray passes through 180 degrees. In other words, the clock goes from 6am to 6pm as the sun moves through the sky.