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

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.

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

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:

• Jesus Christ: The Number of His Name, by Bonnie Gaunt, which contains a riff on the number “37” that John Nash at his looniest would be proud of.

• 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.”

  — Peter J. Boyer in The New Yorker

• 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 —

  19th Nervous Breakdown

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.

Inaugural Poem for California:

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,

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,

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.

The Tables of Time

Implied by previous two entries:

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.

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

“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.

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.