Log24

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: , — m759 @ 4:04 pm

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

The image “http://www.log24.com/log/pix05B/051120-NonEuclideanRev.jpg” cannot be displayed, because it contains errors.

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to  the “Diamond Theory” of truth in The Non-Euclidean Revolution

“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*….”

Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

 In 1490, The White Knight
 (Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

Saturday, November 12, 2005

Saturday November 12, 2005

Filed under: General — Tags: — m759 @ 8:00 pm
Seven is Heaven,
Eight is a Gate


(continued)

A Singer 7-Cycle

“… problems are the poetry of chess.
They demand from the composer
 the same virtues that characterize
all worthwhile art:
originality, invention,
harmony, conciseness,
complexity, and
splendid insincerity.”

Vladimir Nabokov

Saturday, June 11, 2005

Saturday June 11, 2005

Filed under: General — m759 @ 7:10 pm
Picture This

In memory of film producer Fernando Ghia:

“Among Ghia’s solo credits as a producer is

The image “http://www.log24.com/log/pix05A/050611-Lamb3.jpg” cannot be displayed, because it contains errors.

Lady Caroline Lamb,’ a 1972 period drama
written and directed by Robert Bolt.”

Today’s LA Times

Ghia died on June 1, 2005
(the date of the Dutch “No” vote).
In the spirit of Pale Fire, here is an excerpt
from a Log24 entry of that date:

The Road to Brussels


“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”

 — Henry Kissinger, quoted in
     Drama of the Diagonal, Part Deux

Les livres d’histoire et la vie
racontent la même comédie….

Alain Boublil




“Along the road from Ohain to Braine-l’Alleud that hemmed in the plain of Mont-St-Jean and cut at right angles the road to Brussels, which the Emperor wished to take, he [Wellington] had placed 67,000 men and 184 cannons.” Fr. Libert, Waterloo

In researching this entry, I thought of
Wellington’s statement
in “Lady Caroline Lamb” —
These are the Scots Greys.”

The image “http://www.log24.com/log/pix05A/050601-Forever.jpg” cannot be displayed, because it contains errors.

and found the above picture.

Related material:

Women’s History Month.

Wednesday, June 1, 2005

Wednesday June 1, 2005

Filed under: General — Tags: — m759 @ 7:20 pm
The Road to Brussels


“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”

 — Henry Kissinger, quoted in
     Drama of the Diagonal, Part Deux

Les livres d’histoire et la vie
racontent la même comédie….

Alain Boublil


Wellington at Waterloo


The image “http://www.log24.com/log/pix05/050601-Map.jpg” cannot be displayed, because it contains errors.

livgenmi.com/gardiner85.htm 


“Along the road from Ohain to Braine-l’Alleud that hemmed in the plain of Mont-St-Jean and cut at right angles the road to Brussels, which the Emperor wished to take, he [Wellington] had placed 67,000 men and 184 cannons.” Fr. Libert, Waterloo

The Emperor’s Welcome

From Expatriate Online:
Your Bookmark to Belgium

The image “http://www.log24.com/log/pix05/050601-Waterloo.jpg” cannot be displayed, because it contains errors.

Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 pm
Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares
(ps),
a discussion of a 4×4 array
of square linear designs.
  Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle:  the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16.  The first group is the most obvious.  The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
  Being and Nothingness,
  Philosophical Library, 1956
  [reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
 

  The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1. on Page 51:
“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being.  Man is ‘a being of distances.'”
2. on Page 154:
“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities.  Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3. on Page 155:
“… end of the desires.  But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the  glass-in-the-midst-of-the-world.”
4. on Page 158:
“…  it is in time that my possibilities appear on the horizon of the world which they make mine.  If, then, human reality is itself apprehended as temporal….”
5. on Page 180:
“… else time is an illusion and chronology disguises a strictly logical order of  deducibility.  If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6. on Page 186:
“…  It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7. on Page 332:
“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon.  It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8. on Page 359:
“… eyes as objects which manifest the look.  The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9. on Page 392:
“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.  on Page 502:
“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”
11.  on Page 506:
“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.  on Page 661:
“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.  on Page 754:
“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself.  The pen and the pipe, the clothing, the desk, the house– are myself.  The totality of my possessions reflects the totality of my being.  I am what I have.  It is I myself which I touch in this cup, in this trinket.  This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

The image “http://www.log24.com/log/pix05/050527-CipraLogo.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050527-CIPRAview.jpg” cannot be displayed, because it contains errors.
 
From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

Wednesday, March 30, 2005

Wednesday March 30, 2005

Filed under: General — m759 @ 1:28 pm

Logocentric Theology

Logic is all about the entertaining of possibilities.”

— Colin McGinn,
Mindsight: Image, Dream, Meaning,
Harvard U. Press (See yesterday’s entry.)

“God is the sum of all possibilities.”

— Isaac Bashevis Singer, according to
the Associated Press “Today in History
feature for today, March 30, 2005

“A probability space is a measure space with total measure one.”

Gregory F. Lawler, Probability Notes

“Hear, O Israel: The Lord our God, the Lord is one.”

Deuteronomy 6:4

For other illustrations of logocentric theology, see

Matrix of the Death God (May 25, 2003),

Transcendental Meditation (July 30, 2003),

and, for Warren Beatty’s birthday today,

Graphical Password
(April 27, 2003).

Friday, December 3, 2004

Friday December 3, 2004

Filed under: General — m759 @ 2:56 pm

Crimson
on St. Cecilia’s Day

“… from the Age that is past,
To the Age that is waiting before.”
— Samuel Gilman, “Fair Harvard

Published by The Harvard Crimson
on Monday, November 22, 2004:

Dylan Performs
for Sold-Out Crowd

By KATHERINE CHAN
Harvard Crimson Contributing Writer

Shouts of “Make way! Moses is here!” filled a restless crowd as legendary musician Bob Dylan closed off his College tour last night jamming in front of a sold out audience of Harvard undergraduates and Cambridge residents….

The turnout for last night’s two-hour show was greater than many of the student audience members anticipated…

But despite the legendary hits and massive crowds, several students said they were disappointed with the show.

“I love Bob Dylan. I just don’t know what he’s saying,” said Alexander A.C. De Carvalho ’08.

Recommended reading
for Harvard students:

The image “http://www.log24.com/log/pix04B/041203-Lyrics.jpg” cannot be displayed, because it contains errors.

Click on picture
for details.

From an entry of October 29, 2004:

“Each epoch has its singer.”
Jack London, Oakland, California, 1901

“Anything but the void. And so we keep hoping to luck into a winning combination, to tap into a subtle harmony, trying like lock pickers to negotiate a compromise with the ‘mystery tramp,’ as Bob Dylan put it….”
— Dennis Overbye, Quantum Baseball,
    New York Times, Oct.  26, 2004

“You said you’d never compromise
With the mystery tramp,
    but now you realize
He’s not selling any alibis
As you stare into
    the vacuum of his eyes
And ask him do you want to
    make a deal?”
— Bob Dylan, Like a Rolling Stone

From The New York Times today:

“It’s official, I guess. Forty years after he recorded it, Bob Dylan’s ‘Like a Rolling Stone’ was just named the greatest rock ‘n’ roll song of all time….”

Friday, October 29, 2004

Friday October 29, 2004

Filed under: General — m759 @ 12:12 pm

Song

"Each epoch has its singer."
Jack London, Oakland, California, 1901

"Anything but the void. And so we keep hoping to luck into a winning combination, to tap into a subtle harmony, trying like lock pickers to negotiate a compromise with the 'mystery tramp,' as Bob Dylan put it…."
— Dennis Overbye, Quantum Baseball,
    New York Times, Oct.  26, 2004

"You said you'd never compromise
With the mystery tramp,
    but now you realize
He's not selling any alibis
As you stare into
    the vacuum of his eyes
And ask him do you want to
    make a deal?"
— Bob Dylan, Like a Rolling Stone

"About a century ago scientists began to realize that beneath the too, too solid veneer of what had passed for reality for 2,000 years there was some pretty funny and fuzzy business going on….

Most of us, I suspect, would rather believe that the devil is running things than that no one is in charge, that our lives, our loves, World Series victories, hang on the whims of fate and chains of coincidences, on God throwing dice, as Einstein once referred to quantum randomness….

[But] we are people, with desires and memories and a sense of humor – not Ping Pong balls."
— Dennis Overbye, Quantum Baseball,
    New York Times, Oct.  26, 2004

"You can be replaced by some Ping Pong balls and a dictionary."
Anonymous source, March 29, 2001

Friday, August 6, 2004

Friday August 6, 2004

Filed under: General — m759 @ 6:29 pm

Epistle and Hymns

In the spirit of Dave Barry’s Book of Bad Songs, we conclude our Hiroshima Day service with a link to The Epistle of Jeremiah and a deadly trinity of singers:

A Landmark

Neil Diamond–
I Am … I Said

Hoyt Axton–
Jeremiah Was a Bullfrog

Jill O’Hara–
It’s Not Easy Being Green

Monday, April 19, 2004

Monday April 19, 2004

Filed under: General — m759 @ 7:59 pm

Cartesian Theatre

From aldaily.com today:

"If my mind is a tiny theatre I watch in my brain, then there is a tinier mind and theatre inside that mind to see it, and so on forever… more»"

This leads to the dream (or nightmare) of the Cartesian theatre, as pictured by Daniel Dennett.

From websurfing yesterday and today…

The tiny theatre of Ivor Grattan-Guinness:

"… mathematicians often treat history with contempt (unsullied by any practice or even knowledge of it, of course)."

The Rainbow of Mathematics

The contempt for history of the Harvard mathematics department (see previous entry) suggests a phrase….

A search on "Harvard sneer" yields, as the first page found, a memorial to an expert practitioner of the Harvard sneer… Robert Harris Chapman, Professor of English Literature, playwright, theatrical consultant, and founding Director of the Loeb Drama Center from 1960 to 1980.

Continuing the Grattan-Guinness rainbow theme in a tinier theatre, we may picture Chapman's reaction to the current Irish Repertory Theatre production of Finian's Rainbow.  Let us hope it is not a Harvard sneer.

In a yet tinier theatre, we may envision a mathematical version of Finian's Rainbow, with Og the leprechaun played by Andrew P. Ogg.  Ogg would, of course, perform a musical version of his remarks on the Jugendtraum:

"Follow the fellow who follows a dream."

Melissa Errico
in Finian's Rainbow

"Give her a song like…. 'Look to the Rainbow,' and her gleaming soprano effortlessly flies it into the stratosphere where such numbers belong. This is the voice of enchantment…."

Ben Brantley, today's NY Times

For related philosophical remarks on rainbows, infinite regress, and redheads, see

Loretta's Rainbow and

The Leonardo Code.

Sunday, March 28, 2004

Sunday March 28, 2004

Filed under: General — m759 @ 12:12 pm

American Heaven

Headlines from today’s Google News:

Singer Jan Berry, 62; Half of Surf Music Duo

Screeching for heaven at Mach 7

March 25 news story:

“The promise of 70 virgins in paradise and the equivalent of about $20 was all it took to convince a Palestinian teenager to turn himself into a suicide bomber…”

A more modest paradise, from a Jan Berry obituary today:

With Brian Wilson of the Beach Boys, William Jan Berry co-wrote the lyrics for “Surf City” with its lines about taking the station wagon to a place where there are “two girls for every boy.”*

* Theological footnote for feminists:
In some other regions of American Heaven, there may be two boys for every girl.

Sunday, February 8, 2004

Sunday February 8, 2004

Filed under: General,Geometry — m759 @ 2:00 pm

The Quality of Diamond

On February 3, 2004, archivist and abstract painter Ward Jackson died at 75.  From today’s New York Times:

“Inspired by painters like Piet Mondrian and Josef Albers, Mr. Jackson made austere, hard-edged geometric compositions, typically on diamond-shaped canvases.”

On a 2003 exhibit by Pablo Helguera that included Mr. Jackson:

Parallel Lives

Parallel Lives recounts and recontextualizes real episodes from the lives of five disparate individuals including Florence Foster Jenkins, arguably the world’s worst opera singer; Giulio Camillo, a Renaissance mystic who aimed to build a memory container for all things; Friedrich Froebel, the inventor of the kindergarten education system, the members of the last existing Shaker community, and Ward Jackson, the lifelong archivist of the Solomon R. Guggenheim Museum.

Parallel Lives pays homage to Hans-Georg Gadamer (1900-2002) and his system of philosophical hermeneutics built through an exploration of historicity, language, and art. This exhibition, which draws its title from the classic work by Plutarch, is a project that explores biography as a medium, drawing from the earlier innovation of the biographical practice in works like Marcel Schwob’s “Imaginary Lives” (1896) and John Aubrey’s “Brief Lives” (1681). Through display means, the project blends the lives of these individuals into one basic story, visually stating the relationship between individualism and society as best summarized by Gadamer’s famous phrase: “we all are others, and we all are a self.”

On February 3, the day that Jackson died, there were five different log24.net entries:

  1. The Quality with No Name 
  2. Speaking Globally
  3. Lila
  4. Theory of Design
  5. Retiring Faculty.

Parallels with the Helguera exhibit:

Florence Foster Jenkins: Janet Jackson in (2) above.

Giulio Camillo: Myself as compiler of the synchronistic excerpts in (5).

Friedrich Froebel: David Wade in (4).

The last Shakers: Christopher Alexander and his acolytes in (1).

Ward Jackson: On Feb. 3, Jackson became a permanent part of Quality — i.e., Reality — itself, as described in (3).

Some thoughts of Hans-Georg Gadamer
relevant to Jackson’s death:

Gadamer, Art, and Play

by G.T. Karnezis

The pleasure it [art] elicits “is the joy of knowledge.” It does not operate as an enchantment but “a transformation into the true.” Art, then, would seem to be an essentializing agent insofar as it reveals what is essential. Gadamer asks us to see reality as a horizon of “still undecided possibilities,” of unfulfilled expectations, of contingency. If, in a particular case, however, “a meaningful whole completes and fulfills itself in reality,” it is like a drama. If someone sees the whole of reality as a closed circle of meaning” he will be able to speak “of the comedy and tragedy of life” (genres becoming ways of conceiving reality). In such cases where reality “is understood as a play, there emerges the reality of what play is, which we call the play of art.” As such, art is a realization: “By means of it everyone recognizes that that is how things are.” Reality, in this viewpoint, is what has not been transformed. Art is defined as “the raising up of this reality to its truth.”

As noted in entry (3) above
on the day that Jackson died,

“All the world’s a stage.”

William Shakespeare

Tuesday, January 20, 2004

Tuesday January 20, 2004

Filed under: General — m759 @ 1:00 pm

Song of the Father

The death of Max D. Barnes (previous entry) and the opening of the first Tennessee lottery suggested the following meditations.

Wikipedia on Jimmie Rodgers, known as the father of country music:

“Fundamentally, Rodgers was a white blues singer….”

A song by the father of country music:

T for Texas, T for Tennessee,
T for Texas, T for Tennessee,
T for Thelma, that gal
made a wreck out of me.

Gonna buy me a shotgun,
long as I am tall,
Buy me a shotgun,
long as I am tall,
Gonna shoot po’ Thelma,
just to see her jump and fall.

From Wikipedia:

“In modern Western popular music, call and response is most commonly found in the blues and in blues-derived music like jazz and rock’n’roll.”

If Rodgers’s song is the call, what, one wonders, would be the appropriate response?

Sunday, January 11, 2004

Sunday January 11, 2004

Filed under: General — m759 @ 2:11 am

String Theory

Phil Sweetland of the New York Times on Gospel singer St. Jake Hess, who died on Sunday, January 4, 2004 — also the feast day of saints T. S. Eliot, T.S. Matthews, and Joan Aiken

“Mr. Hess was the string that tied together many of Christian music’s most famous quartets and ensembles, and he was an idol and later a colleague of [Elvis] Presley….

Mr. Hess sang at his funeral in 1977, as he had at the funeral of Hank Williams in 1953.”

“Go to other people’s funerals,
otherwise, they won’t go to yours.”

Proverb attributed to St. Yogi Berra

“… to apprehend
 The point of intersection of the timeless
 With time, is an occupation for the saint…. “

— T. S. Eliot, Four Quartets

“You win again.”

— Keith Richards,
tribute to Hank Williams

Sunday, December 7, 2003

Sunday December 7, 2003

Filed under: General — m759 @ 1:11 pm

The Last Samurai
Grandmother

“The ‘Samurai Grandmother’
Margaret Singer
   has passed away…”

Singer was the author of
Cults in Our Midst.

For some background on
Singer and Scientology,
see The Anti-Cult Movement.

” ‘I might look like a little old grandma, but I’m no pushover,’ she told a reporter last year, just before tossing back another shot of Bushmills Irish whiskey, her libation of choice.”

“Occasionally threatened, Singer refused to back down. In a 2002 interview with the San Francisco Chronicle, she told how, at 80, she had frightened off someone who’d been leaving menacing notes in her mailbox.”


I’ve got a 12-gauge shotgun
up here with a spray pattern that’ll put a three-foot hole in you, sonny, and you’d better get off my porch, or you’ll be sorry!” she shouted out the window.

Sunday, November 23, 2003

Sunday November 23, 2003

Filed under: General — m759 @ 11:23 pm

A Contribution to Trudeau’s
Story Theory of Truth” —

Epic of the Chosen People:

After Forty Years
in the Wilderness,

The Winners Are…

Dallas, 1963:

Sam the Sham
“started his music career
in Dallas in the early sixties”
— The Pharaohs Discography

Leesville, La., 2003:

Forty years later,
Leesville to honor
“Wooly Bully” singer

Friday, November 7, 2003

Friday November 7, 2003

Filed under: General — m759 @ 1:28 pm

Today in History:
The Comeback Kid

(Courtesy of Dr. Mac’s Cultural Calendar)

On this date:

In 1962, having lost the California governor’s race, Richard Nixon said to the press, “You won’t have Nixon to kick around any more.”

In 1972, Republican incumbent President Richard Nixon was re-elected, defeating Democratic candidate George McGovern, 520 electoral votes to 17.

From the archives of singer/songwriter Shannon Campbell (“voice of an angel, mouth of a truckdriver”)–

Feb. 6, 2002: The Essential Matrix

NEO: (whines) Who am I?

TRINITY: You are The One.

EVERYONE ELSE: Eh, he might be The One.

TRINITY: He is The One.

NEO: I am not The One.

TRINITY: You are The One.

THE ORACLE: You are not The One, but you can’t tell anybody.

NEO: (whines) But I wanted to be The One. I want to go home….

TRINITY: Fuck. He’s not The One.

EVERYONE ELSE: Told you so.

MORPHEUS: Sure wish someone was The One. I’m in deep shit.

Sunday, October 26, 2003

Sunday October 26, 2003

Filed under: General — m759 @ 3:17 am

ART WARS for

Trotsky’s Birthday

Part I:
Symbols

From my entry of July 26, 2003, in memory
of Marathon Man director John Schlesinger:

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

Dark Lady

For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.

No se puede vivir sin amar.


Part II:
Sunday in the Park with Death

  To Leon from Diego —
Details of a mural,
A Dream of a Sunday Afternoon
in Alameda Park,
Fresco, 1947-48,
Alameda Hotel, Mexico City:

Three’s a Crowd:

Symbol:


Saturday, October 11, 2003

Saturday October 11, 2003

Filed under: General — m759 @ 12:25 am

The Mysterious West

Thanks again to KHYI, Plano, Texas, for great poetry.  In tonight’s KHYI playlist…

From Spike and Jamie:

WAIT A WHILE AND YOU’LL GROW OLDER;
NEVER MIND WHAT THE OLD FOLKS SAY.

JUST KEEP AN ANGEL ON YOUR SHOULDER;
AND NEVER THROW YOUR DREAMS AWAY
FOR THEY MIGHT SAVE YOUR LIFE ONE DAY.

SONG IS JUST A BOX OF VISIONS;
YOU CAN UNLOCK IT WITH A KEY–

A MESSAGE ROLLED UP INSIDE A BOTTLE
AND DROPPED INTO THE SALTY SEA.

SONG IS JUST A BOX OF VISIONS,
A JAR OF HARPS AND GYPSY’S EARS,
 
A LABYRINTH OF WILD ROSES,
A JOURNEY THROUGH A HOUSE OF MIRRORS.

WAIT A WHILE AND YOU’LL GROW STRONGER;
NEVER MIND WHAT THE SAD FOLKS SAY.

From Tish Hinojosa:

It’s the way of life in the real west…”

A search for information on the singer of “Real West” led to a site in Japan that mentions Hinojosa, among many other makers of recommended music:

From Japan–

Random Diary & Essay

“an example of understand beyond language is still possible”

Such an example is one of the themes in a movie I admire greatly….

Ghost Dog – The Way of the Samurai.

The hero’s understanding of what his friend says, even though he does not know the friend’s language, is a recurring theme in this film.

As for me… “No entiendo.  Sigo trabajando.”

Monday, September 29, 2003

Monday September 29, 2003

Filed under: General — m759 @ 3:03 pm

Magic Hawaii

Today, the birthday of singer Jerry Lee Lewis, is also the feast of St. Michael and All Angels.

In honor of Lewis:

Killer Radio, an entry of July 31, 2003, that contains the following…

“When the light came she was sitting on the bed beside an open suitcase, toying with her diamond rings.  She saw the light first in the depths of the largest stone.”

— Paul Preuss, Broken Symmetries,
    scene at Diamond Head, Oahu,
    Hawaii

In honor of the angels:

Mathematics as an Adequate Language,
by Israel Gelfand, Sept. 2, 2003, which contains the following…

“Many people consider mathematics to be a boring and formal science.  However, any really good work in mathematics always has in it: beauty, simplicity, exactness, and crazy ideas.  This is a strange combination.  I understood earlier that this combination is essential on the example of classical music and poetry.  But it is also typical in mathematics.  It is not by chance that many mathematicians enjoy serious music.

This combination of beauty, simplicity, exactness, and crazy ideas is, I think, common to both mathematics and music.”

These qualities seem also to be sought by practitioners of religion and physics… for example, by the spiritually-minded physicist in Preuss’s Broken Symmetries.  Skeptics might prefer, to the word “religion,” the word (pronounced with a sneer) “magic.”

What do we find if, following in the footsteps of Gelfand and Preuss, we do a Google search on the following words…

beauty simplicity exactness
 crazy magic Hawaii
“?

The search yields two results:

  1. The Pupil: Poems by W. S. Merwin.
    The above link is to a poem, “Prophecy,” that seems suitable for these, the High Holy Days at the end of one year and the beginning of another.

    For a follow-up to the poem, see
    The Shining of Lucero.

  2. Striking Through the Mask, or
     The Allegorical Meanings
     in Moby Dick
    .”

These two selections, both on the theme of light and darkness, offer a language that is perhaps more adequate than mathematics for dealing with the nature of the High Holy Days.  For a more lighthearted approach to these concerns, also with a Hawaiian theme, see

The Aloha Mass.

Wednesday, September 24, 2003

Wednesday September 24, 2003

Filed under: General — m759 @ 10:01 pm

Bel Canto

The conclusion of tonight’s season-
opening episode of “The West Wing”
was a picture of President Bartlet  
 receiving the Host at Mass.

Related material:

The Source:

Tips On Popular Singing
by Frank Sinatra
in collaboration with
his vocal teacher John Quinlan

What prompted me to find this
booklet on the Web
(at about 8:45 PM tonight) was

40,000 Years of Music
by Jacques Chailley
(Farrar, Straus & Giroux, 1964),
page 162,
on the bel canto style of singing.

I picked up this book this afternoon
at a sale for $1.

See also Sinatra’s remarks on bel canto
(various places on the Web).

For the religious significance of
the page number 162, see my
entry of 9/11 2003,

Particularity.


Added at 3:20 AM Sept. 25…

In Related News:

Source: Google News, about 3:15 AM 9/25/03

Friday, September 12, 2003

Friday September 12, 2003

Filed under: General — Tags: — m759 @ 3:33 pm

Into the Sunset

I just learned of Johnny’s Cash’s death.  On Google News, the headline was  Johnny Cash rides into sunset.  The source was the Bangkok Post.

“Don’t you know that
when you play at this level
there’s no ordinary venue.”

One Night in Bangkok (midi)



No Ordinary Venue

“They are the horses of a dream.
 They are not what they seem.”

The Hex Witch of Seldom, page 16

A Singer 7-Cycle
A Singer
7-Cycle

The Magnificent Seven:

CLICK HERE for 

“the adventures of filming this epic
on location in Cuernavaca, Mexico.”

“He is the outlaw the people love,
the hero dressed in black.”

The Hex Witch of Seldom,
by Nancy Springer, page 15

“Words are events.”

Walter J. Ong, Society of Jesus 

“…search for thirty-three and three…”
The Black Queen in The Eight

Friday September 12, 2003

Filed under: General — Tags: — m759 @ 3:06 pm

Commentary
on the two previous entries

On 4:04:08:

“Je ne connais que deux sortes d’êtres immuables sur la terre: les géomètres et les animaux; ils sont conduits par deux règles invariables la démonstration et l’instinct; et encore les géomètres ont-ils eu quelques disputes, mais les animaux n’ont jamais varié.”

— Voltaire, Dictionnaire Philosophique, “Des Contradictions dans les Affaires et dans les Hommes

A Singer 7-cycle

 On 4:04:08
and on
Particularity:

“El pan que se come no es pan.”

— Voltaire quoting Montesquieu
on the Pope’s declarations,
Spanish translation

Saturday, August 2, 2003

Saturday August 2, 2003

Filed under: General — m759 @ 7:59 am

Dark Desire

Film star dies after fight
with rock boyfriend

“…they seemed destined to become France’s golden couple: the fragile and gifted film actress from one of the country’s great theatrical families, and the radical rock star-poet with a genuine social conscience.

But yesterday Marie Trintignant died in Paris of a cerebral haemorrhage, while her boyfriend, Bernard [sic] Cantat, lead singer of France’s most popular rock band

Noir Désir

was in jail… suspected of landing the blow that plunged her into a coma from which she never emerged.

— Jon Henley in Paris
    Saturday August 2, 2003
   
The Guardian 

The Details:

“Trintignant… was rushed to hospital at 7.30 on Sunday morning…. 

The singer, adored in France as much for his militant and public stands on issues such as racism, globalisation and the Israeli-Palestinian dispute as for his powerful lyrics and charismatic stage presence, was admitted to hospital shortly afterwards with acute alcohol poisoning and a suspected overdose of prescription drugs.

He had allegedly waited more than five hours since the midnight struggle before sounding the alarm….”

Last Sunday’s site music, for the entry Catholic Tastes, was…  

Nous Voici Dans La Ville – A Christmas song from 15th century France (midi by John Philip Dimick).  

It will serve as a memorial song for Marie.

As for Cantat, see the 
four entries that preceded
the Catholic Tastes entry.

These deal with substance abuse and postmodern French philosophy.

The song I would recommend to memorialize the role of Cantat in this affair is American rather than French…

 

Pukin’ in the Parkin’ Lot.”

Religious meditation for today:

As remarked in my
obituary for Sam Phillips,
Father of Rock and Roll,

“If there’s a rock and roll heaven,
Well you know they’ve got
a hell of a band.”

Tuesday, July 29, 2003

Tuesday July 29, 2003

Filed under: General — m759 @ 10:00 am

Trick of the Light

For Carly Simon

“… on the dance floor she seemed to be the only one completely alive.  It was a trick of the light that followed one person around.  Joe had seen the quality before; it was rare, but not unknown.

Every time we say good-bye…. Porter had written an intimate ballad…. “

— Martin Cruz Smith, Stallion Gate, Ch. 2

“At night I heard God
                          whisper lullabyes
While Daddy next door
                   whistled whisky tunes
And sometimes
            when I wanted,
                   they would harmonize
There was nothing
                    those two couldn’t do

……………………………………………

Then one night Daddy died
                      and went to Heaven
And God came down to earth
                          and slipped away
I pretended not to notice
                      I’d been abandoned
But no-one sang the night
                                 into the day
And later night time songs
                          came back again
But the singers don’t compare
                        with those I knew
And I never figured out
          where God and Daddy went
But there was nothing
                   those two couldn’t do”

— Carly Simon,
  “Embrace Me, You Child”

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General — Tags: — m759 @ 5:29 am

Funeral March

John Schlesinger dead at 77;
‘Midnight Cowboy’ director

 
Anthony Breznican
Associated Press
Jul. 26, 2003 12:00 AM

LOS ANGELES – Oscar-winning director John Schlesinger, who daringly brought gay characters into mainstream cinema with Midnight Cowboy and tapped into nightmares with the teeth-drilling torture of Marathon Man, died Friday at 77.

The British-born filmmaker…. died about 5:30 a.m….

Schlesinger also directed The Day of the Locust, based on a novel by Nathanael West.

See Heaven, Hell, and Hollywood and

Dogma Part II: Amores Perros.

From the latter:

“Then you know your body’s sent,
Don’t care if you don’t pay rent,
Sky is high and so am I,
If you’re a viper — a vi-paah.”

The Day of the Locust,
    by Nathanael West (1939),
    New Directions paperback,
    1969, page 162

This song may be downloaded at

Pot Culture, 1910-1960.

That same site begins with a traditional Mexican song…

La cucaracha, la cucaracha,
 ya no puede caminar,
 porque no quiere,
 porque le falta
 marihuana que fumar.
” 

(“The cockroach, the cockroach,
 can’t walk anymore,
 because he doesn’t want to,
 because he has no
 marihuana to smoke.”)

This suggests an appropriate funeral march for John Schlesinger:

“Ya murió la cucaracha, ya la llevan a enterrar…”La Cucaracha

Those attending Schlesinger’s wake, as opposed to his funeral, may wish to perform other numbers from the Pot Culture page, which offers a variety of “viper” songs.

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

 

Dark Lady

For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.

No se puede vivir sin amar.

Concluding Unscientific Postscript:

Oh, yes… the question of
Heaven or Hell for John Schlesinger… 

Recall that he also directed the delightful
Cold Comfort Farm and see
last year’s entry for this date.

Friday, June 27, 2003

Friday June 27, 2003

Filed under: General — m759 @ 4:05 pm

Countin’ Flowers on the Wall

Today, the birthday of Bob Keeshan (aka Captain Kangaroo), seems an appropriate day to meditate on religious issues.

The death of minimalist artist Fred Sandback on June 23, coupled with my vitriolic attack on Christianity of that same date, suggest that I should give some recognition to defenders of that bizarre religion.

Two of Christianity’s most able defenders are singer Martina McBride and professor of English Gerald McDaniel.

McBride has stated loudly and clearly her conviction that

Love’s the only house big enough for all the pain in the world.”

I may be mistaken, but this sounds to me like a profession of faith in the Christian church.

McDaniel’s Cultural Calendar includes many items of Christian interest (saints’ days, etc.) but also includes a greater variety of general cultural history than any other online calendar I know of.  Like all Christian documents, it displays little regard for the truth (dates are often wrong), but its heart seems to be in the right place.  Most important, in light of McBride’s persuasive song lyric, is the fact that McDaniel’s calendar — and therefore, perhaps, his church — does seem to be big enough — at least in principle — for some of the pain in the world, as well as some of the joy.  Unfortunately, the church has much less room for truth than for emotion.

For emotion, see McBride’s album of that title.  For some thoughts on truth, see McBride’s fellow country singer, Patty Loveless (whom I greatly prefer to McBride).

The above reflection was prompted by McDaniel’s having a written a document that includes both of the following items for June 27:

  • TV’s Captain Kangaroo, Bob Keeshan, was born on this day in 1927. He was also Clarabelle the Clown on Howdy Doody.
  • On this day in 1928, an American expatriate bookseller, Sylvia Beach, and her companion Adrienne Monnier, had a dinner party to which they invited James Joyce and Scott Fitzgerald. It was held at her flat above her bookstore Shakespeare & Company….

These two items appear to have been placed by McDaniel in the right date slot.  (Christians may sometimes have the right values, but should never be trusted to have their facts straight.)

Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 12:07 am

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets.  It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."


H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

 
The "horse's head" figure above is from a note I wrote on this date 18 years ago.  The following journal entry from April 4, 2003, gives some details:
 

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.

Tuesday, April 22, 2003

Tuesday April 22, 2003

Filed under: General — m759 @ 4:23 pm

Temptation


Locomotive

The Star
of Venus


Locomotion

In memory of Nina Simone, a singer who died April 21, whose autobiography was titled (after the Screamin’ Jay Hawkins song) I Put a Spell on You, and in honor of Aaron Spelling, producer of “Satan’s School for Girls,” whose birthday is today, I suggest the following three cultural milestones.

First, an accurate, if tasteless, recounting of Scripture at a Christian site that correctly notes that Satan may appear as “an angel of light”… rather like Aaron Spelling?  This site also offers, as background music, a lame parody of the evils of Rock ‘n’ Roll in the form of a midi of “Fire” that would hardly tempt even someone Hell-bent on sinning. 

Second, a book, The Club Dumas, by Arturo Perez-Reverte, the basis of the Roman Polanski film “The Ninth Gate.”  This book is notable for the way it skillfully, and perhaps accurately, depicts Satan as an “angel of light” who does not resemble Aaron Spelling in the least.   This Satan could really tempt me.

Finally, my favorite music video of all time: the 1988 Kylie Minogue “Locomotion.”  If the Devil could now look, sing, and dance like Kylie in 1988, I would be lost.  Fortunately, perhaps, the days when Kylie could make me fall in love with one glance are now over.  Still, if I had to fall, I would much rather do it with Kylie than with Spelling.  As she herself says,  

It’s better the Devil you know.”

For more on Kylie, trains, and death, see the Jan. 3 entry The Shanghai Gesture

The locomotive image is courtesy of a website that may, in view of the subject of this entry, prefer to remain anonymous.

Saturday, April 5, 2003

Saturday April 5, 2003

Filed under: General,Geometry — Tags: — m759 @ 9:49 am

Art Wars:
Mathematics and the
Emperor's New Art

From Maureen Dowd's New York Times column of June 9, 2002: 

"The shape of the government is not as important as the policy of the government. If he makes the policy aggressive and pre-emptive, the president can conduct the war on terror from the National Gallery of Art."

 

NY Times, April 5, 2003:
U.S. Tanks Move Into Center of Baghdad
See also today's op-ed piece
by Patton's grandson.

Meanwhile, at the Washington Post, another example of great determination and strength of character:

 

Donald Coxeter Dies: Leader in Geometry

By Martin Weil
Washington Post Staff Writer
Saturday, April 5, 2003

"Donald Coxeter, 96, a mathematician who was one of the 20th century's foremost specialists in geometry and a man of great determination and strength of character as well, died March 31 at his home in Toronto."

From another Coxeter obituary:

In the Second World War, Coxeter was asked by the American government to work in Washington as a code-breaker. He accepted, but then backed out, partly because of his pacifist views and partly for aesthetic reasons: "The work didn't really appeal to me," he explained; "it was a different sort of mathematics."

For a differing account of how geometry is related to code-breaking, see the "Singer 7-cycle" link in yesterday's entry, "The Eight," of 3:33 PM.  This leads to a site titled

An Introduction to the
Applications of Geometry in Cryptography
.

"Now I have precisely the right instrument, at precisely the right moment of history, in exactly the right place."

 — "Patton,"
the film

Quod erat
demonstrandum
.


 

Added Sunday, April 6, 2003, 3:17 PM:

The New York Times Magazine of April 6
continues this Art Wars theme.


                 (Cover typography revised)

The military nature of our Art Wars theme appears in the Times's choice of words for its cover headline: "The Greatest Generation." (This headline appears in the paper, but not the Internet, version.)

Some remarks in today's Times Magazine article seem especially relevant to my journal entry for Michelangelo's birthday, March 6.

"…Conceptualism — suddenly art could be nothing more than an idea….

LeWitt moved between his syntax of geometric sculptures and mental propositions for images: concepts he wrote on paper that could be realized by him or someone else or not at all.  Physical things are perishable.  Ideas need not be."

— Michael Kimmelman, chief art critic of the New York Times, April 6, 2003

Compare this with a mathematician's aesthetics:

"A mathematician, like a painter or a poet, is a maker of patterns.  If his patterns are more permanent than theirs, it is because they are made with ideas."

— G. H. Hardy, A Mathematician's Apology (1940), reprinted 1969, Cambridge U. Press, p. 84 

It seems clear from these two quotations that the real conceptual art is mathematics and that Kimmelman is peddling the emperor's new clothes.

Friday, April 4, 2003

Friday April 4, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 3:33 pm

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.


Decimal 
labeling


Binary
labeling


Algebraic
labeling

IMAGE- Knight figure for April 4
Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.  (See the 4:36 PM entry.)

 

 

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.
 

Monday, October 14, 2002

Monday October 14, 2002

Filed under: General — m759 @ 11:11 pm

Going His Way

October 14 in history:

1888 Katherine Mansfield, author, is born.

1977 Bing Crosby, singer/actor (Going My Way), dies.

“He was given up to his dream. What did garden-parties and baskets and lace frocks matter to him? He was far from all those things. He was wonderful, beautiful…. Happy … happy … All is well, said that sleeping face. This is just as it should be. I am content.”

— Katherine Mansfield, “The Garden Party

In honor of Mansfield, Crosby, and other authors and singers, this site’s music is now a midi rendition of Rick Nelson’s classic.

Thursday, September 19, 2002

Thursday September 19, 2002

Filed under: General,Geometry — m759 @ 2:16 pm

Fermat’s Sombrero

Mexican singer Vincente Fernandez holds up the Latin Grammy award (L) for Best Ranchero Album he won for “Mas Con El Numero Uno” and the Latin Grammy Legend award at the third annual Latin Grammy Awards September 18, 2002 in Hollywood. REUTERS/Adrees Latif

From a (paper) journal note of January 5, 2002:

Princeton Alumni Weekly 
January 24, 2001 

The Sound of Math:
Turning a mathematical theorem
 and proof into a musical

How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor? 

 

Wallace Stevens:
Poet of the American Imagination

Consider these lines from
“Six Significant Landscapes” part VI:

Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses-
As, for example, the ellipse of the half-moon-
Rationalists would wear sombreros.

Addendum of 9/19/02: See also footnote 25 in

Theological Method and Imagination

by Julian N. Hartt

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