The Proof and the Lie

A mathematical lie has been circulating on the Internet.

It concerns the background of Wiles's recent work on mathematics related to Fermat's last theorem, which involves the earlier work of a mathematician named Taniyama.

This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as "elliptic curves" and "modular forms."

The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, "The Proof," in October 1997 and repeated in a book based on the program and in a Scientific American article, "Fermat's Last Stand," by Simon Singh and Kenneth Ribet, in November 1997.

"... elliptic curves and modular forms... are from opposite ends of the mathematical spectrum, and had previously been studied in isolation."

-- Site on Simon Singh's 1997 book Fermat's Last Theorem

"JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that's so hard to explain.

BARRY MAZUR: So, let me explain.  Over here, you have the elliptic world, the elliptic curves, these doughnuts.  And over here, you have the modular world, modular forms with their many, many symmetries.  The Shimura-Taniyama conjecture makes a bridge between these two worlds.  These worlds live on different planets.  It's a bridge.  It's more than a bridge; it's really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.

KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous...."

-- Transcript of NOVA program, "The Proof," October 1997

The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:

"Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn't believe that the two are related."

-- Ivars Peterson, "Curving Beyond Fermat," November 1999 

The lie has now contaminated university mathematics courses, as well as popular accounts:

"Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together."

-- Site on Fermat's last theorem by undergraduate K. V. Binns

Authors like Singh who wrote about Wiles's work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat's last theorem.

A 1996 page in Daney's site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms.  Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.

Here is what Daney says:

"Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/r -> C.  j is therefore an example of what is called a modular function. We'll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions."

"Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996"

Update of Dec. 2, 2003

For the relationship between modular functions and modular forms, see (for instance) Modular Form in Wikipedia.

Some other relevant quotations:

From J. S. Milne, Modular Functions and Modular Forms:

"The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenth-century] theory of elliptic functions."

The next quote, also in a nineteenth-century context, relates elliptic functions to elliptic curves.

From Elliptic Functions, a course syllabus:

"Elliptic functions parametrize elliptic curves."

Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama's 1955 conjecture), between elliptic curves and modular forms.

Another quote from Milne, to summarize:

"From this [a discussion of nineteenth-century mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms."

Serge Lang apparently agrees:

"Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century."

-- Editorial description of Lang's Elliptic Functions (second edition, 1987)

Update of Dec. 3, 2003

"The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal."

-- Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp.  (The review  appeared in Bulletin of the American Mathematical Society, January 1994.)

Sermon for St. Andrew's Day:

Necessity Is the Mother of Invention.

(Flag thanks to a Stevenson family site)

Command at Mount Sinai

Tuesday, Nov. 25, was the feast day of St. Catherine, patroness of a monastery at Mount Sinai. (See entries for that date.)

"In a landmark essay,* the anthropologist Bernard S. Cohn showed how the command of language† could become the language of command.‡"

-- "Right formula for a nation in the making,"
     by Asad Latif

* "The Command of Language
    and the Language of Command,"
    Subaltern Studies IV, pp. 276- 329

† B.S. "I think writing about people in science and math is a way we can pay homage to genius and people we admire.  And it's a way of saying, 'You may be smarter, but I have the last word, I control you.' " -- Ira Hauptman, author of a play, "Partition," about the mathematician Ramanujan and the culture of India

‡ No B.S. NY Times, Saturday, Nov. 29, 2003: B. S. Cohn, Expert on Culture of Modern India, Dies at 75  CHICAGO, Nov. 28 — Bernard S. Cohn, who spent his life studying and writing about British influence on modern Indian culture and society, died here on Tuesday....

Understanding Media

BBC News, March 2, 2002:

"The Reverend Billy Graham has apologised for a taped conversation with former President Nixon in which he said the Jewish 'stranglehold' of the media was ruining the United States and must be broken."

"The ‘propaganda model’ of media operations laid out and applied by Edward Herman and Noam Chomsky in Manufacturing Consent: The Political Economy of the Mass Media postulates that elite media interlock with other institutional sectors in ownership, management and social circles, effectively circumscribing their ability to remain analytically detached from other dominant institutional sectors. The model argues that the net result of this is self-censorship without any significant coercion." 

-- A Critical Review and Assessment of
   Herman and Chomsky’s ‘Propaganda Model'
   by Jeffery Klaehn,
   European Journal of Communication,
   2002, Vol 17(2): 147–182. 

"We are in a war of ideas."

-- Defense Secretary Donald Rumsfeld,
   Oct. 24, 2003

Toby Ziegler
of West Wing

THE WEST WING
"NIGHT FIVE"

WRITTEN BY: AARON SORKIN

Transcription from http://communicationsoffice.tripod.com
/3-13.txt

Episode 3.13 -- "Night Five"
Original Airdate: February 6, 2002

**************************

PORTION OF THE TRANSCRIPT
DEALING WITH A
UN FOREIGN POLICY SPEECH

**************************

TOBY
Our goal is to proclaim American values.

ANDY
This speech isn't supposed to be about ideology. It's supposed to be about reality.

TOBY
I think the President will decide what the speech is suppose to be about, but the reality is, the United States of America no longer sucks up to reactionaries, and our staunch allies will know what we mean.

ANDY
We don't have any staunch allies in the Arab world; just reluctant ones.  We've a coalition held together with duct tape! A coalition without which we cannot fight!

TOBY
Nobody's blowing off the coalition, and that coalition will be plenty strong....

ANDY
What's Egypt going to think? Or Pakistan?

TOBY
That freedom and democracy are coming soon to a theatre near them, so get dressed.

He sits on the edge of his desk.

ANDY
.... this one moment in time, you have to
get off your horse and just... simply put - be nice to the Arab world.

TOBY
Be nice?

ANDY
Yes.

TOBY
Well... How about when we, instead of
blowing Iraq back to the seventh century for harbouring terrorists and trying to develop nuclear weapons,
we just imposed economic sanctions and were reviled by the Arab world....

Supplemental reading:

Who Rules America?

Review of Abraham Foxman's
Never Again? The Threat of
The New Anti-Semitism,
NY Times Book Review,
November 30, 2003

Wheels for St. Catherine

"This java applet displays the wave functions of a particle in a three dimensional harmonic oscillator."

See also the Chapel of the Burning Bush at St. Catherine's Monastery.

Bad Manners

Paul Krugman in today's New York Times:

"One of the problems with media coverage of this administration," wrote Eric Alterman in The Nation, "is that it requires bad manners."

He's right. There's no nice way to explain how the administration uses cooked numbers to sell its tax cuts, or how its arrogance and gullibility led to the current mess in Iraq.

The bad manners that are required go far beyond "media coverage of this administration."

The arrogance and gullibility of George W. Bush have two roots, both religious:

• The arrogance of the Jews, reflected in that contemptible pack of Jewish lies, the Old Testament, and
• The gullibility of the Christians, reflected in the more recent damned Jewish lies of the New Testament.

As an example of the bad manners required, read the obituary of a great man, Edward L. Schempp, who died on Nov. 8, 2003.

His efforts ended, in 1963, a Pennsylvania state law that required Bible reading in schools.

The generation that passed that state law is now also, for the most part, dead.  Good riddance.

The only way to stop such religious insanity is to repudiate the intolerance of our own parents, grandparents, great-grandparents, and so on.  If this means admitting that we were raised by lying assholes, so be it.

Of course, those politically opposed to the religious right are also, for the most part, lying assholes.*  In fact, this phrase pretty much sums up the whole damned human race.  (With a few exceptions, some of them named Catherine... See previous entry.)

Have a nice day.

* See, for instance, the following statement by playwright Ira Hauptman:

"I think writing about people in science and math is a way we can pay homage to genius and people we admire. And it's a way of saying, 'You may be smarter, but I have the last word, I control you.' And, beyond that, I think there is a connection between science and math and playwrights; we're all creating imaginary universes."

St. Catherine's Day

As the previous three entries indicate, I have little respect for the lies of the Bible.  Certain Christian traditions are, however, worthy of respect.... among them, the observance of Nov. 25 as St. Catherine's Day.

Farewell to 40-Year Holiday

Hope everyone had a happy "Sam the Sham Day," a religious holiday with roots in the Book of Exodus:

"We got the... name from the movie 'The Ten Commandments.' Old Ramses, the King of Egypt, looked pretty cool, so we decided to become The Pharaohs."

-- Sam the Sham

(See also previous two entries.)

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

For Saint Cecilia's Day

"Bastian Perrot... constructed a frame, modeled on a child's abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time values of the notes, and so on. ... What later evolved out of that students' sport and Perrot's bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game."

-- Hermann Hesse, Das Glasperlenspiel

Compare and contrast:

School Book Depository

"Many people look at the Kennedy assassination as a turning point, when people started realizing and thinking and believing their government would lie to them and lie to them repeatedly," said Gary Mack, curator of the Sixth Floor Museum at Dealey Plaza in Dallas.

-- AP, Dallas, Nov. 21, 2003

Better late than never.

Chinese Theatre, Part II:

Just Say NO

For more on the above "spider" symbol, see

ART WARS for Trotsky's Birthday
(Oct. 26, 2003), Parts I and II

and the site from which
the above figure is taken,

Yin & Yang and the I Ching.

For some Chinese poetic justice, see

The Song of Saint Ezra,

Library of Paradise, and

Endings and Beginnings.

See, too, the Chinese character for "end"
used to sell the work of Ian Fleming:

Note, in Endings and Beginnings, the strong resemblance between this character and the name of the Chinese-American architect of the Robert Frost Library at Amherst College.  Then meditate on the following passage by Amherst graduate Stephen Mitchell:

“We dance round in a ring and suppose,
But the Secret sits in the middle
    and knows,”
Robert Frost wrote,
looking in from the outside.
Looking out from the inside,
    Chuang-tzu wrote,
“When we understand, we are at
    the center of the circle,
and there we sit while Yes and No
    chase each other
around the circumference.”

A view of the Robert Frost Library
from the inside is available in the entry

Library of Paradise

mentioned above.

See, too, my entry

Keats and the Web

of July 28, 2002.

November Oscar

From weebay.com:

November This nautical flag signifies the letter N.	Oscar This nautical flag signifies the letter O.

Just say November Oscar.

(See previous entry.)

Chinese Theatre

Epic Records released a new Michael Jackson album, "Number Ones," on Tuesday, Nov. 18, 2003.

From Those Were the Days:

On this date in...

"1984 - The largest crowd to see the unveiling of a Hollywood Walk-of-Fame star turned out as Michael Jackson got his piece of the sidewalk right in front of Mann’s Chinese Theatre in Los Angeles.

1971 - Isaac Hayes of Memphis, TN, got his first #1 hit as the 'Theme from Shaft' began a two-week stay at the top of the charts."

The above two Chinese characters
mean "Shaft One," according to

The Source:

Unihan 3.1 data for U+4E95,

Unihan 3.1 data for U+4E28.

Hayes won an Oscar for best song.

Staying the Course

"However flawed the case for invading Iraq may have been, the premature withdrawal of U.S. military forces would not only be a humiliating defeat for the United States but a betrayal of the hopes of the Iraqi people...."

-- "Staying the Course," editorial in America, the Jesuit weekly, Nov. 24, 2003

"...all means to prevent procreation are illicit. This includes temporary or permanent sterilization, chemicals (like birth control pills or foams), mechanical devices (like the condom or diaphragm) or premature withdrawal."

-- "The Wisdom of Humanae Vitae," by Father Jay Scott Newman

"This is a perfect example of what my father calls 'thinking with your dick.' "

-- Susanna Moore, author of In the Cut

Today's birthday: Meg Ryan, star of the film version of In the Cut.

See also the previous entry.

Total Recall:

in which Philip K. Dick
meets Joan Didion yet again

From Joan Didion's new work on California history, Where I Was From:

"There was never just the golden dream of riches and bountiful nature, but always a scene of exploitation and false promises, indifference and ruthlessness, a kind of hollow core."

Hollow no more.      

Archaischer Torso Apollos

by Rainer Maria Rilke

Wir kannten nicht sein unerhörtes Haupt,
darin die Augenäpfel reiften. Aber
sein Torso glüht noch wie ein Kandelaber,
in dem sein Schauen, nur zurückgeschraubt,

sich hält und glänzt. Sonst könnte nicht der Bug
der Brust dich blenden, und im leisen Drehen
der Lenden könnte nicht ein Lächeln gehen
zu jener Mitte, die die Zeugung trug.

Sonst stünde dieser Stein entstellt und kurz
unter der Schultern durchsichtigem Sturz
und flimmerte nicht so wie Raubtierfelle;

und bräche nicht aus allen seinen Rändern
aus wie ein Stern: denn da ist keine Stelle,
die dich nicht sieht. Du musst dein Leben ändern.

Illustration:

See also

Philip K. Dick Meets Joan Didion,

Aes Triplex,

From the Empty Center,

The Empty Center, and

Translation of Rilke by Stephen Mitchell:

Archaic Torso of Apollo

We cannot know his legendary head
with eyes like ripening fruit. And yet his torso
is still suffused with brilliance from inside,
like a lamp, in which his gaze, now turned to low,

gleams in all its power. Otherwise
the curved breast could not dazzle you so, nor could
a smile run through the placid hips and thighs
to that dark center where procreation flared.

Otherwise this stone would seem defaced
beneath the translucent cascade of the shoulders
and would not glisten like a wild beast's fur:

would not, from all the borders of itself,
burst like a star: for here there is no place
that does not see you. You must change your life.

Russell Crowe as Santa's Helper

From The Age, Nov. 17, 2003:

"Russell Crowe's period naval epic has been relegated to second place at the US box office by an elf raised by Santa's helpers at the North Pole."

From A Midsummer Night's Dream:

"The lunatic,¹ the lover,² and the poet³
  Are of imagination all compact."

In acceping a British Film Award for his work in A Beautiful Mind, Crowe said that

"Richard Harris, one of the finest of this profession, recently brought to my attention the verse of Patrick Kavanagh:

'To be a poet and not know the trade,
To be a lover and repel all women,
Twin ironies by which
    great saints are made,
The agonising
    pincer jaws of heaven.' "

A theological image both more pleasant and more in keeping with the mathematical background of A Beautiful Mind is the following:

This picture, from a site titled Strange and Complex, illustrates a one-to-one correspondence between the points of the complex plane and all the points of the sphere except for the North Pole.

To complete the correspondence (to, in Shakespeare's words, make the sphere's image "all compact"), we may adjoin a "point at infinity" to the plane -- the image, under the revised correspondence, of the North Pole.

For related poetry, see Stevens's "A Primitive Like an Orb."

For more on the point at infinity, see the conclusion of Midsummer Eve's Dream.

For Crowe's role as Santa's helper, consider how he has helped make known the poetry of Patrick Kavanagh, and see Kavanagh's "Advent":

O after Christmas† we'll have
    no need to go searching....

... Christ comes with a January flower.

† i.e. Christ Mass... as, for instance, performed by the six Jesuits who were murdered in El Salvador on this date in 1989.
  

The Empty Center

From Stephen Mitchell, foreword to
The Enlightened Heart:
An Anthology of Sacred Poetry:

“We dance round in a ring and suppose,
But the Secret sits in the middle
    and knows,”
Robert Frost wrote,
looking in from the outside.
Looking out from the inside,
    Chuang-tzu wrote,
“When we understand, we are at
    the center of the circle,
and there we sit while Yes and No
    chase each other
around the circumference.”
This anonymous center—
which is called God
in Jewish, Christian, and Moslem cultures,
and Tao, Self, or Buddha
in the great Eastern Traditions—
is the realest of realities.

From Wallace Stevens's
A Primitive Like an Orb:

The essential poem
    at the center of things....

We do not prove
    the existence of the poem.
It is something seen and known
    in lesser poems.

From Namkaran Samskar:

There is only one center in existence;
the ancients used to call it
Tao, Dharma, God.

Those words have become old now;
you can call it Truth.

There is only one center of existence.

From the
Empty Center:

Today's previous entry, "Aes Triplex," is actually from 1:48 PM EST yesterday.  (It was posted to my alternate site, log24.com, since log24.net was down for Xanga maintenance.)  "Aes Triplex" deals with image and reality.

Its final link, to the heart of Rome, leads to Julius Caesar.

A related review in today's New York Times:

The opening paragraph:

"We live in a media maelstrom, and the Moonwork theater company's 'Julius Caesar' comes hurtling toward us right from its center. This production, at the Connelly Theater in the East Village through Nov. 23, is set in the here and now.

Shakespeare's

'Julius Caesar' is about politics, rhetoric and power; about manipulation of a nation's image and its people; about conspiracy, murder and the war that leads to a new regime. What play is better suited for our times?"

-- Margo Jefferson

Aes Triplex

The title, from a Robert Louis Stevenson essay, means "triple brass" (or triple bronze):

From the admirable site of J. Nathan Matias:

"Aes Triplex means Triple Bronze, from a line in Horace's Odes that reads 'Oak and triple bronze encompassed the breast of him who first entrusted his frail craft to the wild sea.' ''

From Philip K. Dick's The Man in the High Castle:

Juliana said, "Oracle, why did you write The Grasshopper Lies Heavy? What are we supposed to learn?"

"You have a disconcertingly superstitious way of phrasing your question," Hawthorne said. But he had squatted down to witness the coin throwing. "Go ahead," he said; he handed her three Chinese brass coins with holes in the center. "I generally use these." 

This passage, included in my earlier entry of Friday, combined with the opening of yet another major motion picture starring Russell Crowe, suggests three readings for that young man, who is perhaps the true successor to Marlon Brando.

Oracle, for Crowe as John Nash (A Beautiful Mind):

Understanding the I Ching

Mutiny, for Crowe as Jack Aubrey (Master and Commander):

Bartleby, the Scrivener

Storm, for Crowe as Maximus (Gladiator):

Pharsalia, Book V:
The Oracle, the Mutiny, the Storm

As background listening, one possibility is Sinatra's classic "Three Coins":

"Three hearts in the fountain,
Each heart longing for its home.
There they lie in the fountain
Somewhere in the heart of Rome.*" 

Personally, though, I prefer, as a tribute to author Joan Didion (who also wrote of coins and the Book of Transformations), the even more classic Sinatra ballad

"Angel Eyes."

 * Horace leads to "Acroceraunian shoals," which leads to Palaeste, which leads to Pharsalia and to the heart of Rome.  (With a nod to my high school Latin teacher, the late great John Stachowiak.)

Philip K. Dick Meets Joan Didion

From the ending of
The Man in the High Castle:

Juliana said, "I wonder why the oracle would write a novel. Did you ever think of asking it that?" ....

"You may say the question aloud," Hawthorne said. "We have no secrets here."

Juliana said, "Oracle, why did you write The Grasshopper Lies Heavy? What are we supposed to learn?"

"You have a disconcertingly superstitious way of phrasing your question," Hawthorne said. But he had squatted down to witness the coin throwing. "Go ahead," he said; he handed her three Chinese brass coins with holes in the center. "I generally use these." 

She began throwing the coins; she felt calm and very much herself. Hawthorne wrote down her lines for her. When she had thrown the coins six times, he gazed down and said:

"Sun at the top. Tui at the bottom. Empty in the center."

"Do you know what hexagram that is?" she said. "Without using the chart?"

"Yes," Hawthorne said.

"It's Chung Fu," Juliana said. "Inner Truth. I know without using the chart, too. And I know what it means."

From the ending of
Play It As It Lays:

I lie here in the sunlight, watch the hummingbird.  This morning I threw the coins in the swimming pool, and they gleamed and turned in the water in such a way that I was almost moved to read them.  I refrained.

One thing in my defense, not that it matters.  I know something Carter never knew, or Helene, or maybe you.  I know what "nothing" means, and keep on playing.

Why, BZ would say.

Why not, I say.

The Tables of Time

Implied by previous two entries:

"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond,

Is immortal diamond."

-- Gerard Manley Hopkins,

"That Nature is a Heraclitean Fire

and of the Comfort of the Resurrection" New York Times, Nov. 13, 2003:

See, too, Art Wars and Time Fold.

Change and Permanence:
A Philosophical Quartet

Dream of the Unified Field

Quartet:

Shanavasa, Ananda,
Jorie Graham, Robert Louis Stevenson 

"Shanavasa asked Ananda,

'What is the fundamental uncreated essence of all things?' "

-- Jorie Graham,
    "Relativity: A Quartet"
    in The Dream of the Unified Field:
    Selected Poems 1974-1994,
    Ecco Press, 1995

"Ananda to Shanavasa:
 'Buddha is Alive! Buddha is Alive!'

 Shanavasa to Upagupta:
 'Space is Consumed by Flaming Space.' "

 -- Table of Contents, Living Buddha Zen

Cover illustration
by Stephen Savage,
NY Times Book Review,
Feb. 2 (Candlemas), 2003

"We live the time that a match flickers."

-- Robert Louis Stevenson, Aes Triplex

Robert Louis Stevenson was born in Edinburgh on this date in 1850.

The Silver Table

"And suddenly all was changed.  I saw a great assembly of gigantic forms all motionless, all in deepest silence, standing forever about a little silver table and looking upon it.  And on the table there were little figures like chessmen who went to and fro doing this and that.  And I knew that each chessman was the idolum or puppet representative of some one of the great presences that stood by.  And the acts and motions of each chessman were a moving portrait, a mimicry or pantomine, which delineated the inmost nature of his giant master.  And these chessmen are men and women as they appear to themselves and to one another in this world.  And the silver table is Time.  And those who stand and watch are the immortal souls of those same men and women.  Then vertigo and terror seized me and, clutching at my Teacher, I said, 'Is that the truth?....' "

-- C.S. Lewis, The Great Divorce, final chapter

Follow-up to the previous four entries:

St. Art Carney, whom we may imagine to be a passenger on the heavenly bus in The Great Divorce, died on Sunday, Nov. 9, 2003.

The entry for that date (Weyl's birthday) asks for the order of the automorphism group of a 4x4 array.  For a generalization to an 8x8 array -- i.e., a chessboard -- see

Geometry of the I Ching.

Audrey Meadows, said to have been the youngest daughter of her family, was born in Wuchang, China. 

"Tui means to 'give joy.'  Tui leads the common folk and with joy they forget their toil and even their fear of death. She is sometimes also called a sorceress because of her association with the gathering yin energy of approaching winter.  She is a symbol of the West and autumn, the place and time of death."

-- Paraphrase of Book III, Commentaries of Wilhelm/Baynes.

Divine Comedy

Michael Joseph Gross:

"The Great Divorce is C.S. Lewis's Divine Comedy: the narrator bears strong resemblance to Lewis (by way of Dante); his Virgil is the fantasy writer George MacDonald; and upon boarding a bus in a nondescript neighborhood, the narrator is taken to Heaven...."

11:11

"Why do we remember the past
but not the future?"

-- Stephen Hawking,
A Brief History of Time,
Ch. 9, "The Arrow of Time"

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

"Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right - past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time."

Is Time Quantized?

Yes.

Maybe.

We don't really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional.  Then the following picture,

from Time Fold, of "four quartets" time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday's entry, on the symmetry group of a generic 4x4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot's own brief history of time, Four Quartets:

"It seems, as one becomes older,
That the past has another pattern,
    and ceases to be a mere sequence....

I sometimes wonder if that is
    what Krishna meant—
Among other things—or one way
    of putting the same thing:
That the future is a faded song,
    a Royal Rose or a lavender spray
Of wistful regret for those who are
    not yet here to regret,
Pressed between yellow leaves
    of a book that has never been opened.
And the way up is the way down,
    the way forward is the way back."

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

For Hermann Weyl's Birthday:

A Structure-Endowed Entity

"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."

-- Hermann Weyl in Symmetry

Exercise:  Apply Weyl's lesson to the following "structure-endowed entity."

What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array's structure you choose to examine.  It could be in the hundreds, or in the hundreds of thousands.)

A Beautiful Fantasy:

The Secret life of
 John Nash

"Dr. Blind (pronounced 'Blend') was about ninety years old and had taught, for the past fifty years, a course called 'Invariant Subspaces' which was noted for its monotony and virtually absolute unintelligibility, as well as for the fact that the final exam, as long as anyone could remember, had consisted of the same single yes-or-no question. The question was three pages long but the answer was always 'Yes'. That was all you needed to pass Invariant Subspaces."

-- The Secret History, by Donna Tartt

"...I put my arms around him yes and drew him down to me so he could feel my breasts all perfume yes and his heart was going like mad and yes I said yes I will Yes.

Trieste-Zurich-Paris
1914-1921"

-- Ulysses, by James Joyce

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.

Legacy Codes:

The Most Violent Poem

Lore of the Manhattan Project:

From The Trinity Site --

"I imagined Oppenheimer saying aloud, 'Batter my heart, three person'd God," unexpectedly recalling John Donne's 'Holy Sonnet [14],' and then he knew, ' "Trinity" will do.' Memory has its reasons.

'Batter my heart' -- I remember these words. I first heard them on a fall day at Duke University in 1963. Inside a classroom twelve of us were seated around a long seminar table listening to Reynolds Price recite this holy sonnet....

I remember Reynolds saying, slowly, carefully, 'This is the most violent poem in the English language.' "

Related Entertainment

Today's birthday:
director Mike Nichols

From a dead Righteous Brother:

"If you believe in forever
Then life is just a one-night stand."

-- Bobby Hatfield, found dead
in his hotel room at
7 PM EST Wednesday, Nov. 5, 2003,
before a concert scheduled at
Western Michigan University, Kalamazoo.

From a review of The Matrix Revolutions:

"You'd have to be totally blind at the end to miss the Christian symbolism.... Trinity gets a glimpse of heaven.... And in the end... God Put A Rainbow In The Clouds."

Moral of the
Entertainment:

According to Chu Hsi [Zhu Xi],

"Li" is
"the principle or coherence
or order or pattern
underlying the cosmos."

-- Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990

Related Non-Entertainment

Symmetry and a Trinity
(for the dotting-the-eye symbol above)

Introduction to Harmonic Analysis
(for musical and historical background)

Mathematical Proofs
(for the spirit of Western Michigan
University, Kalamazoo)

Moral of the
Non-Entertainment:

"Many kinds of entity
become easier to handle
by decomposing them into
components belonging to spaces
invariant under specified symmetries."

-- The importance of
mathematical conceptualisation
by David Corfield,
Department of History and
Philosophy of Science,
University of Cambridge

See, too,
Symmetry of Walsh Functions and
Geometry of the I Ching.

Legacy Codes

"In writing The Legacy Codes, the term itself became the play's central metaphor. In newspaper accounts of the Wen Ho Lee case, the classified legacy codes which caused the uproar were described as computer simulations of plutonium explosions. The term is also used by computer experts for any archaic codes which are still necessary to run complex computer programs. For me the term can also be interpreted as the DNA genetic code, it can be interpreted as what is passed on in families regarding culture, family secrets, genetic traits. It also can relate to how people and institutions want to be remembered in the future."

-- Playwright Cherylene Lee

The Legacy Codes opens at 7 tonight in Manhattan.

Game Over

 "Everything that has a beginning
     has an end."

-- The Matrix Revolutions

Matrix, by Knots, Inc., 1979.

"Easy to master -- A lifetime to enjoy!"

The object for 2 players (8-adult)
is to be the first to form a line
consisting of 4 different
colored chips.

Imagist Poem

(Recall the Go-chip
in Wild Palms.)

Endings and Beginnings

Today's birthday: author Sam Shepard.

From pbs.org:

"Shepard has a noted aversion to pat endings: 'I hate endings. Just detest them. . . . The temptation toward resolution, towards wrapping up the package, seems to me a terrible trap. Why not be more honest with the moment? The most authentic endings are the ones which are already evolving towards another beginning. That's genius.' "

Chinese Poem

Translation:

End

Chu

(See previous entry.
Click on characters
above for details.)

Library of Paradise

Click to enlarge.

In memory of architect Philip Chu, who designed the above library at Amherst College:

"Chu was best known for his designs of college libraries, which his family said blended 'modern influences from such innovators as Frank Lloyd Wright, the Oriental use of space and exterior design together with the traditional materials.' Critics characterized his designs as 'warm and inviting,' his family said in a written statement.

Among his designs were the Robert Frost Library at Amherst College, which was dedicated by President John Kennedy..."

-- Honolulu Advertiser, Nov. 3, 2003

And now I was beginning to surmise:
Here was the library of Paradise.

-- Hermann Hesse, Magister Ludi

Chu died at 83 in Honolulu on
October 27, 2003.

See Dream of Heaven, Oct. 27, 2003.

See, too, ART WARS for Oct. 26, 2003...
Forty years to the day after Kennedy's remarks at Amherst.

Chronotope

The polytope at left in the illustration below might, to use the term of Bakhtin, be called a "chronotope."

See Time Fold for a literary context.

All Souls' Day
at the Still Point

From remarks on Denis Donoghue's Speaking of Beauty in the New York Review of Books, issue dated Nov. 20, 2003, page 48:

"The Russian theorist Bakhtin lends his august authority to what Donoghue's lively conversation has been saying, or implying, all along.  'Beauty does not know itself; it cannot found and validate itself — it simply is.' "

From The Bakhtin Circle:

"Goethe's imagination was fundamentally chronotopic, he visualised time in space:

Time and space merge ... into an inseparable unity ... a definite and absolutely concrete locality serves at the starting point for the creative imagination... this is a piece of human history, historical time condensed into space....

Dostoevskii... sought to present the voices of his era in a 'pure simultaneity' unrivalled since Dante. In contradistinction to that of Goethe this chronotope was one of visualising relations in terms of space not time and this leads to a philosophical bent that is distinctly messianic:

Only such things as can conceivably be linked at a single point in time are essential and are incorporated into Dostoevskii's world; such things can be carried over into eternity, for in eternity, according to Dostoevskii, all is simultaneous, everything coexists.... "

Bakhtin's notion of a "chronotope" was rather poorly defined.  For a geometric structure that might well be called by this name, see Poetry's Bones and Time Fold.  For a similar, but somewhat simpler, structure, see Balanchine's Birthday.

From Four Quartets:

"At the still point, there the dance is."

From an essay by William H. Gass on Malcolm Lowry's classic novel Under the Volcano:

"There is no o'clock in a cantina."

Symmetry in Diamond Theory:
Robbing Peter to Pay Paul

"Groups arise in most areas of pure and applied mathematics, usually as a set of operators or transformations of some structure. The appearance of a group generally reflects some kind of symmetry in the object under study, and such symmetry may be considered one of the fundamental notions of mathematics."

-- Peter Webb

"Counter-change is sometimes known as Robbing Peter to Pay Paul."

-- Helen Kelley Patchwork

For a look at the Soviet approach
to counterchange symmetry, see

The Kishinev School of Discrete Geometry.

The larger cultural context:

See War of Ideas (Oct. 24),
The Hunt for Red October (Oct. 25),
On the Left (Oct. 25), and
ART WARS for Trotsky's Birthday (Oct. 26).

All Saints' Day:
The Song of Saint Ezra

Ezra Pound, imagist poet and fascist saint, died on this date in 1972.

 "But you, newest song of the lot,
  You are not old enough
     to have done much mischief.
  I will get you a green coat out of China
  With dragons worked upon it."

-- "Further Instructions," 1913

For more on China and Christian Fascism, see the memorial to the wife of Generalissimo Chiang Kai-shek in TIME magazine, issue dated Nov. 3, 2003.

From Image in Poetry:

"Ezra Pound made perhaps the most widely used definition of image in the 20th century:

An ‘Image’ is that which presents an intellectual and emotional complex in an instant of time. "

-- Ezra Pound, "A Few Don’ts by an Imagiste," Poetry, March 1913

For an excellent essay by Jungian James Hillman on the political implications of imagism, see

Egalitarian Typologies versus
the Perception of the Unique.

A specific image that is a personal favorite of mine is found in the I Ching:

Note that in the West,
this Chinese character
is known as the "Pound sign."

"The Perception of the Unique," indeed.