Sunday, November 30, 2003 3:27 PM
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 Taniyama's conjecture 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"
Sunday, November 30, 2003 12:00 AM
Sermon for St. Andrew's Day:
Necessity Is the Mother of Invention.
(Flag thanks to a Stevenson family site)
Saturday, November 29, 2003 3:24 PM
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
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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 -- Ira Hauptman, author of a play, "Partition," about the mathematician Ramanujan and the culture of India |
No B.S. NY Times, Saturday, B. S. Cohn, 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.... |
Friday, November 28, 2003 3:31 PM
Understanding Media
"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.
-- Defense Secretary Donald Rumsfeld,
Oct. 24, 2003
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
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PORTION OF THE TRANSCRIPT
DEALING WITH A
UN FOREIGN POLICY SPEECH
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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:
Review of Abraham Foxman's
Never Again? The Threat of
The New Anti-Semitism,
NY Times Book Review,
November 30, 2003
Tuesday, November 25, 2003 11:59 PM
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.
Tuesday, November 25, 2003 12:00 AM
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.
Monday, November 24, 2003 11:59 PM
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.)
Sunday, November 23, 2003 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...
Sam the Sham
"started his music career
in Dallas in the early sixties"
-- The Pharaohs Discography
Forty years later,
Leesville to honor
"Wooly Bully" singer
Sunday, November 23, 2003 1:06 PM
Epiphany
Yesterday, to give thanks for the winning score in the Harvard-Yale game (Harvard won, 37 to 19), I browsed the Net to find the religious significance, if any, of the number "37." I encountered the picture at left below, of a burning bush. (It was frame 37 in a sequence of frames from an episode of The Simpsons.) The finger of flame did not seem to lead to anything meaningful, so I ignored it. (Frame 38 in the sequence seems to be a Simpsons version of Edward G. Robinson in "The Ten Commandments.") Then today, lo and behold, the Commandments themselves appeared before my very eyes, as yet another cartoon... this time, on the editorial page of my local paper (reprinted from McKee in the Augusta Chronicle). Combining the two cartoons, we see the Flaming Finger of God in action.
The above thought process is, of course, less than mentally healthy, but may be of anecdotal interest to some. Several other examples of religious insanity seem relevant:
Mel Gibson on his new film, "The Passion" --
When Gibson was in Rome shooting the film, he told an Italian interviewer that he had felt moved by God's spirit to undertake the project. I asked him what he'd meant by that. How did he know that God wanted him to make “The Passion”?
“There are signals,” he said. “You get signals. Signs. ‘Signal graces,’ they're called. It's like traffic lights. It's as clear as a traffic light. Bing! I mean, it just grabs you and you know you have to listen to that and you have to follow it."
Finally, for religious enthusiasts who, like our President, have a Yale background, an article on the mystical properties of Yale's score yesterday in the big game --
Saturday, November 22, 2003 2:45 PM
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:
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Friday, November 21, 2003 7:04 PM
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.
Better late than never.
Friday, November 21, 2003 2:17 PM
Just Say NO
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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,
For some Chinese poetic justice, see
Library of Paradise, and
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
mentioned above.
See, too, my entry
of July 28, 2002.
Friday, November 21, 2003 1:00 AM
November Oscar
From weebay.com:
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November This nautical |
Oscar This nautical |
(See previous entry.)
Thursday, November 20, 2003 2:35 PM
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:
Hayes won an Oscar for best song.
Wednesday, November 19, 2003 12:25 PM
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.
Monday, November 17, 2003 10:49 AM
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."
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Hollow no more.
Monday, November 17, 2003 9:25 AM
Inaugural Poem for California: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,
The Empty Center, and
Translation of Rilke by Stephen Mitchell:
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.
Sunday, November 16, 2003 7:59 PM
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."
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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.
Sunday, November 16, 2003 2:45 PM
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.
