For the "Aleph" of the title in the seal of
the College of the Desert, see the final post
in a search for the College in this journal.
A better-known Aleph appears in a story by Borges.
See Borges + Aleph in this journal.
For the "Zahir" of the title, see another story by Borges
and the coin scenes in the films "No Country for Old Men"
(2007) and "Mojave" (2015).
The word "Zahir" has appeared previously in this journal
in a post of January 11, 2011, Soul and Spirit.
See Coxeter + Aleph in this journal.
Epigraph to "The Aleph," a 1945 story by Borges:
"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2
Three links with a Borges flavor—
Related material
The 236 in yesterday evening's NY lottery may be
viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.
A new illustration for that topic—
This shows a reconcilation of the triples described by Sloane
in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.
Note that the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of Sloane's 83 configuration—
The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 1984-2009.
"Quizá yo acabe por gastar el Zahir a fuerza de pensarlo
y de repensarlo, quizá detrás de la moneda esté Dios."
Related images . . .
See Borges + Aleph in this journal.
* Phrase from a poem by Wallace Stevens
Rachel Dodes in The Wall Street Journal
on All Souls' Day, 2012—
"In one of the first lines uttered by Daniel Day-Lewis, playing Abraham Lincoln in the new Steven Spielberg film opening Nov. 9, he says, 'I could be bounded in a nutshell, and count myself a king of infinite space— were it not that I have bad dreams.'
The line was ripped straight from 'Hamlet,' by Lincoln's favorite writer, William Shakespeare. Tony Kushner, the Pulitzer Prize-winning playwright ('Angels in America') who wrote the script for the film, says that Shakespeare, much like Lincoln, 'had extraordinary mastery over the darkest parts of the human spirit.'"
The above quotation omits Shakespeare's words prefacing the nutshell part— "O God."
These same words in a different tongue— "Hey Ram"— have often been quoted as the last words of Gandhi. (See yesterday's noon post.)
"… for the Highest Essence (brahman ),
which is the core of the world, is identical
with the Highest Self (ātman ), the kernel
of man's existence."
— Heinrich Zimmer, Myths and Symbols
in Indian Art and Civilization , Pantheon
Books, 1946, page 142
Related material: A post linked to here on Friday night
that itself links to a different Shakespeare speech.
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."
— Rosalind Krauss in "Grids,"
October (Summer 1979), 9: 50-64.
Counterexample—
The Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
| Mathematics and Narrative, Illustrated |
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Mathematics |
 Narrative |
Or: Starting Out in the Evening, continued from noon yesterday
Yesterday evening's New York Lottery numbers were 510 and 5256.
For the former, see post 510, Music for Patricias.
For the latter, see Richard Feynman at the Caltech YMCA Lunch Forum on 5/2/56—
"The Relation of Science and Religion."
Some background….
The Aleph
"As is well known, the Aleph is the first letter of the Hebrew alphabet.
Its use for the strange sphere in my story may not be accidental.
For the Kabbala, the letter stands for the En Soph ,
the pure and boundless godhead; it is also said that it takes
the shape of a man pointing to both heaven and earth…."
— Borges, "The Aleph," quoted in Ayn Sof (January 7th, 2011)
The Y
See "Pythagorean Letter" in this journal.
Edenville
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space…. Am I walking into eternity along Sandymount strand?"
(A continuation of this morning's Coxeter and the Aleph)
"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"
| Mathematics and Narrative, Illustrated | |
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Narrative |
"As is well known, the Aleph is the first letter of the Hebrew alphabet.
Its use for the strange sphere in my story may not be accidental.
For the Kabbala, the letter stands for the En Soph ,
the pure and boundless godhead; it is also said that it takes
the shape of a man pointing to both heaven and earth, in order to show
that the lower world is the map and mirror of the higher; for Cantor's
Mengenlehre , it is the symbol of transfinite numbers,
of which any part is as great as the whole."
— Borges, "The Aleph"
From WorldLingo.com —
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"Infinite Jest… now stands as the principal contender
for what serious literature can aspire to
in the late twentieth and early twenty-first centuries."
— All Things Shining, a work of pop philosophy published January 4th
"You're gonna need a bigger boat." — Roy Scheider in "Jaws"
"We're gonna need more holy water." — "Season of the Witch," a film opening tonight
See also, with respect to David Foster Wallace, infinity, nihilism,
and the above reading of "Ayn Sof" as "nothingness,"
the quotations compiled as "Is Nothing Sacred?"
"Whence God is to be understood as a sort of mirror in which all things succeeding one another in the whole course of time have images shining back, a mirror indeed directly beholding itself and all the images existing in it."
— Peter de Rivo, quoted in Wild Materialism , p. 102 (below)
"The stakes were high…. …the status, indeed, the ownership, of logic, a term common to the Faculties of Theology and Art, was in dispute…."
— Wild Materialism: The Ethic of Terror and the Modern Republic , by Jacques Lezra, Fordham University Press, 2010, chapter on "The Logic of Sovereignty," page 102
See also Borges on the Aleph.
"The Faithful who gather at the mosque of Amr, in Cairo,
are acquainted with the fact that the entire universe
lies inside one of the stone pillars that ring its central court…"
"Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar"
Star Wars
In memory of Melvin J. Lasky, editor, 1958-1990, of the CIA-funded journal Encounter:
“Once called as lively, and as bitchy, as a literary cocktail party, Encounter published articles of unrivalled authority on politics, history and literature.”
Lasky died on Wednesday, May 19, 2004. From a journal entry of my own on that date:
This newly-digitized diagram is from a
paper journal note of October 21, 1999.
Note that the diagram’s overall form is that of an eight-point star. Here is an excerpt from a Fritz Leiber story dealing with such a star, the symbol of a fictional organization:
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Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, “Look, Buster, do you want to live?” Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question. The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar. An X superimposed on a plus sign. It looked permanent. … “Here is how it stacks up: You’ve bought your way with something other than money into an organization of which I am an agent….” “It’s a very big organization,” she went on, as if warning me. “Call it an empire or a power if you like. So far as you are concerned, it has always existed and always will exist. It has agents everywhere, literally. Space and time are no barriers to it. Its purpose, so far as you will ever be able to know it, is to change, for its own aggrandizement, not only the present and the future, but also the past. It is a ruthlessly competitive organization and is merciless to its employees.” “I. G. Farben?” I asked grabbing nervously and clumsily at humor. She didn’t rebuke my flippancy, but said, “And it isn’t the Communist Party or the Ku Klux Klan, or the Avenging Angels or the Black Hand, either, though its enemies give it a nastier name.” “Which is?” I asked. “The Spiders,” she said. That word gave me the shudders, coming so suddenly. I expected the sigil to step off her forehead and scuttle down her face and leap at me—something like that. She watched me. “You might call it the Double Cross,” she suggested, “if that seems better.” — Fritz Leiber, |
From last year’s entry,
Indiana Jones and the Hidden Coffer,
of 6/14:
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From Borges’s “The Aleph“:
From The Hunchback of Notre Dame:
Lena Olin and Harrison Ford |
Finally, from an excellent site
on the Knights Templar,
a quotation from Umberto Eco:
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When all the archetypes burst out shamelessly, we plumb the depths of Homeric profundity. Two cliches make us laugh but a hundred cliches move us because we sense dimly that the cliches are talking among themselves, celebrating a reunion . . . Just as the extreme of pain meets sensual pleasure, and the extreme of perversion borders on mystical energy, so too the extreme of banality allows us to catch a glimpse of the Sublime. — “Casablanca: Cult Movies and Intertextual Collage” (1984) from Travels in Hyperreality |
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Indiana Jones |
In memory of Bernard Williams,
Oxford philosopher, who died Tuesday, June 10, 2003.
“…in… Truth and Truthfulness [September, 2002], he sought to speak plainly, and took on the post-modern, politically correct notion that truth is merely relative…”
“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….
A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory….”
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987
Today is the feast day of Saint Jorge Luis Borges (b. Buenos Aires, August 24, 1899 – d. Geneva, June 14, 1986).
From Borges’s “The Aleph“:
“The Faithful who gather at the mosque of Amr, in Cairo, are acquainted with the fact that the entire universe lies inside one of the stone pillars that ring its central court…. The mosque dates from the seventh century; the pillars come from other temples of pre-Islamic religions…. Does this Aleph exist in the heart of a stone?”
(“Los fieles que concurren a la mezquita de Amr, en el Cairo, saben muy bien que el universo está en el interior de una de las columnas de piedra que rodean el patio central…. la mezquita data del siglo VII; las columnas proceden de otros templos de religiones anteislámicas…. ¿Existe ese Aleph en lo íntimo de una piedra?”)
From The Hunchback of Notre Dame:
Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar.*
* A coffer of great richness
In a pillar’s heart they found,
Within it lay new banners,
With figures to astound.
See also the figures obtained by coloring and permuting parts of the above religious symbol.
Lena Olin and Harrison Ford
in “Hollywood Homicide“
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