Wednesday, January 14, 2004  3:00 AM

Language Game

Ludwig Wittgenstein,
Philosophical Investigations:

373. Grammar tells what kind of object anything is. (Theology as grammar.)

Related material:

See this date last year, and

Zen and Language Games

(May 2, 2003).

See also the phrase "May 2, 373."

Wednesday, January 14, 2004  2:00 AM

Games

On this date —

Alfred Tarski was born
in 1902 in Warsaw, and

Kurt Friedrich Gödel died
in 1978 in Princeton.

From last year's entry on this date:

What is Truth?

"What is called 'losing' in chess
may constitute winning
in another game."

-- Ludwig Wittgenstein,
Remarks on the
Foundations of Mathematics
(revised edition, MIT Press, 1978)

Tuesday, January 13, 2004  11:45 PM

At Last, Some Veritas

From the Harvard Crimson, 1/12/04:

College Faces Mental Health Crisis

"An overwhelming majority of Harvard undergraduates struggle with mental health problems, a recent Crimson poll found."

Related material:

"The people who intermediate between lunatics and the world used to be called alienists; the go-betweens for mathematicians are called teachers. Many a student may rightly have wondered if the terms shouldn't be reversed."

-- Book review in the current Harvard Magazine; among the authors reviewed is Harvard mathematician and administrator Benedict H. Gross.

"Dean of the College Benedict H. Gross ’71 has said improving mental health services is one of his top priorities in his first year on the job."

-- Harvard Crimson 1/12/04

"He takes us to the central activity of mathematics—which is imagining...."

-- Harvard Magazine on Harvard mathematician and author Barry Mazur.

For related material on Mazur, see

A Mathematical Lie.

"The teenagers aren't all bad. I love 'em if nobody else does. There ain't nothing wrong with young people. Jus' quit lyin' to 'em."

-- Jackie "Moms" Mabley

Tuesday, January 13, 2004  4:15 PM

Deeply Deep

"Remember your epiphanies on green oval leaves, deeply deep, copies to be sent if you died to all the great libraries of the world, including Alexandria?"

-- James Joyce, Ulysses, "Proteus"

James Joyce may or may not have been a saint.  Today is, accordingly, either his feast day or his secular day of remembrance.

With Joyce in mind, I surfed the Heckler & Coch weblog archives this afternoon and found a link to a page that credits

"Jørn Barger, an amateur
  James Joyce scholar...."

with the first use of the term "weblog" in its current sense.

Seeking more on Barger and Joyce, I found that Barger has gone into seclusion and that his Joyce website is no longer online.

Google has a cache of his Joyce portal, however, and the portal and its sub-pages are also available at the Internet Archive Wayback Machine:

http://web.archive.org/web/2003*/
http://www.robotwisdom.com/jaj/*

These pages from Barger's labor of love, though neither green nor oval, may serve as this year's Joyce memorial.

Sunday, January 11, 2004  1:11 PM

In Summary

To sum up the last two entries:

"I returned and saw under the sun
 that the race is not to the swift,
 nor the battle to the strong,
 nor bread to the wise,
 nor riches to men of understanding,
 nor favor to men of skill;
 but time and chance
 happeneth to them all"

-- Ecclesiastes 9:11.

Sunday, January 11, 2004  1:00 PM

Two-Dimensional Time

The following is from the Prime Quotes page at the website of Matthew R. Watkins...

"I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time."

J.J. Sylvester, from "On certain inequalities relating to prime numbers", Nature 38 (1888) 259-262, and reproduced in Collected Mathematical Papers, Volume 4, page 600 (Chelsea, New York, 1973)

Translated into contemporary English, Sylvester is saying more-or-less this:

"I have sometimes thought that if we were able to perceive time in some multi-dimensional way, more like a surface than like a line, then perhaps the distribution of prime numbers would be entirely self-evident, and would not seem at all mysterious to us."

Many thanks to Heckler & Coch (5/19/03) for pointing out the Sylvester quotation.

For related thoughts on this topic, see Time Fold.

Sunday, January 11, 2004  11:11 AM

The Lottery

What these numbers mean to me:

720: See the recent entries

Music for Dunne's Wake,

720 in the Book, and

Report to the Joint Mathematics Meetings.

616 and 201:

The dates, 6/16 and 2/01,
of Bloomsday and St. Bridget's Day.

510:  A more difficult association...

Perhaps "Love at the Five and Dime"
(8/3/03 and 1/4/04).

Perhaps Fred Astaire's birthday, 5/10.

More interesting...

A search for relevant material in my own archives, using the phrase "may 10" cullinane journal, leads to the very interesting weblog Heckler & Coch, which contains the following brief entries (from May 19, 2003):

"May you live in interesting times
While widely reported as being an ancient Chinese curse, this phrase is likely to be of recent and western origin.

Geometry of the I Ching
The Cullinane sequence of the 64 hexagrams"

"... there are many associations of ideas which do not correspond to any actual connection of cause and effect in the world of phenomena...."

-- John Fiske, "The Primeval Ghost-World," quoted in the Heckler & Coch weblog

"The association is the idea"

-- Ian Lee on the communion of saints and the association of ideas (in The Third Word War, 1978)

Sunday, January 11, 2004  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

Friday, January 9, 2004  7:20 AM

Report to the
Joint Mathematics Meetings

"What was the lecture about,
Cosmo wanted to know.

'It's about solving equations
of the fifth degree,
which are supposed to be insoluble.'"

-- Chapter 2 of
The Shadow Guests,
by Joan Aiken

For more material on insolubility
of fifth-degree equations
and on this winter's
Joint Mathematics Meetings
(Phoenix, Jan. 7-10), see
the January 6 entry
720 in the Book.

For more material on Joan Aiken,
who died on January 4,
see the previous entry.

The number 720 is the order of
the symmetric group of degree 6.

For material related to
exceptional outer automorphisms
of this group and to
a song about Arizona, see

Skewed Mirrors.

Arizona Star:

"Shinin' like a diamond
 she had tombstones
in her eyes."

Friday, January 9, 2004  2:01 AM

HURRY UP PLEASE
IT'S TIME

-- T. S. Eliot,
The Waste Land, II
"A Game of Chess"

Jan. 9 obituary of Brian Gibson --

"In 2002 he was executive producer of the film 'Frida,' about the artist Frida Kahlo...."

Captured for the Queen

Joan Aiken

Photo by Alex Gotfryd,
circa 1972 

Jan. 9 obituary of Joan Aiken --

"Joan Aiken was born in Rye, England, a daughter of the American poet Conrad Aiken...."

Dust jacket of a novel -- 

"Malcolm Lowry's Under the Volcano must be, for anyone who loves the English language, a sheer joy."

-- Conrad Aiken

"He was never inclined to small talk."

-- Jan. 9 obituary of Steven Edward Dorfman, writer of questions (i.e., answers) for the game show "Jeopardy!"

"What's the Hellfire Club?"

-- Joan Aiken, beginning of the final chapter of The Shadow Guests

Note that Dorfman, Gibson, and Aiken
all died on Sunday, Jan. 4, 2004.
For some related material, see

Sunday in the Park with Death.

Thursday, January 8, 2004  4:23 PM

Natasha's Dance

It seems, according to Eliot's criterion, that the late author John Gregory Dunne may be a saint.

Pursuing further information on the modular group, a topic on which I did a web page Dec. 30, 2003, the date of Dunne's death, I came across a review of Apostol's work on that subject (i.e., the modular group, not Dunne's death, although there is a connection).  The review:

"A clean, elegant,
absolutely lovely text..."

Searching further at Amazon for a newer edition of the Apostol text, I entered the search phrase "Apostol modular functions" and got a list that included the following as number four:

Natasha's Dance:
A Cultural History of Russia,

which, by coincidence, includes all three words of the search.

For a connection -- purely subjective and coincidental, of course -- with Dunne's death, see The Dark Lady (Jan. 1, 2004), which concerns another Natasha... the actress Natalie Wood, the subject of an essay ("Star!") by Dunne in the current issue of the New York Review of Books.

The Review's archives offer another essay, on science and religion, that includes the following relevant questions:

"Have the gates of death
been opened unto thee?
Or hast thou seen the doors
of the shadow of death?"

From my December 31 entry:

In memory of
John Gregory Dunne,
who died on
Dec. 30, 2003:

For further details, click
on the black monolith.

Tuesday, January 6, 2004  10:10 PM

720 in the Book

Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.

An article titled On Mathematical Imagination concludes by looking forward to

"a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation...."

Seeking the source of the phrase "new instauration," I found it was due to Francis Bacon, who "conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of 'the seal of God on things,' " according to a web page by Nieves Mathews.

Hmm.

The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:

Abel's Proof:  An Essay
on the Sources and Meaning
of Mathematical Unsolvability

by Peter Pesic,
MIT Press, 2003

From a review:

"... the book is about the idea that polynomial equations in general cannot be solved exactly in radicals....

Pesic concludes his account after Abel and Galois... and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations.  The reader is left with little clarity on this sequel to the story...."

-- Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244

Here, it seems, is my epiphany:

"Elliptic modular functions suffice to solve all polynomial equations."

Incidental Remarks
on Synchronicity,
Part I

Those who seek a star
on this Feast of the Epiphany
may click here.

Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.

Just how such equations can be solved is a less familiar story.  I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.

The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions.  Modular functions are also distantly related, via the topic of "moonshine" and via the "Happy Family" of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4x4 matrices.

Incidental Remarks
on Synchronicity,
Part II

There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.

Here is what I was able to find on the Web about Pesic's claim:

From Wolfram Research:

From Solving the Quintic --

"Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein's approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions."

From Siegel Theta Function --

"Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)"

From Polynomial --

"... the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable.  Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron.  Klein's method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or 'Siegel functions' must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be 'natural' generalizations of the elliptic functions."

Belardinelli, G. "Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales." Mémoral des Sci. Math. 145, 1960.

King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.

Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440-448, 1999. 

From Angel Zhivkov,

Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:

"... discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker:  in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function....  Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist.  This hypothesis was realized in [a] few cases by F. Klein... Jordan... showed that any algebraic equation is solvable by modular functions.  In 1984 Umemura realized the Kronecker idea in his appendix to Mumford's book... deducing from a formula of Thomae... a root of [an] arbitrary algebraic equation by Siegel modular forms."  

-- "Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants"

Incidental Remarks
on Synchronicity,
Part III

From Music for Dunne's Wake:

"Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged."

— Carrie Fisher,
Postcards from the Edge

"720 in   the Book" and "Paradise"

"The group Sp₄(F₂) has order 720,"
as does S₆. -- Angel Zhivkov, op. cit.

Those seeking
"a rediscovery of
'the seal of God on things,' "
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.

For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see

Log24, June 2003.

For the relevance of the time
of this entry, 10:10, see

Related recreational reading:

Tuesday, January 6, 2004  6:23 AM

"Shining Forth"

Sunday, January 4, 2004  10:10 PM

Room 1010

Continuing the hotel theme of the previous entry....

John Gregory Dunne has a letter in the New York Review of Books of December 20 (St. Emil's Day in the previous entry), 1990.  In this letter, he reveals that he and his wife had at one time worked on a Grand Hotel screenplay based in Las Vegas.

For related material in memory of Dunne, see In Lieu of Rosebud, which contains entries for 10/10-10/12, 2002.

Mein Irisch Kind,
Wo weilest du?

Sunday, January 4, 2004  2:17 PM

2:17

"... both a new world
And the old made explicit, understood
In the completion of its partial ecstasy,
The resolution of its partial horror."

-- T. S. Eliot, Four Quartets 

Speaking of horror, today's noon entry has a link to a page that references Stephen King's The Shining.

On a 1970's edition of
Stephen King's The Shining:

"The page where Danny actually enters room 217 for the first time (King builds to this moment for a long time, it's one of the more frightening passages in the book), is precisely on page 217. Scared the crap out of me the first time I read it."

In honor of St. Thomas Stearns Eliot, whose feast day is today, of St. Emil Artin (see entries for St. Emil's day, 12/20/03), and of Room 217, a check of last year's 2/17 entries leads to St. Andrea's weblog, which today, recalling the "white and geometric" prewar Berlin of the 12/20/03 entries, has Andrea looking, with Euclid, on beauty bare.

See also my entry "The Boys from Uruguay" and the later entry "Lichtung!" on the Deutsche Schule Montevideo in Uruguay.

Sunday, January 4, 2004  12:00 PM

Noon

"These fragments I have shored against my ruins" -- T. S. Eliot, The Waste Land

Friday, January 2, 2004  4:28 PM

Music for Dunne's Wake

"Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged."

 — Carrie Fisher,
Postcards from the Edge

"720 in   the Book" and "Paradise"

Musical Note: A Star is Born

Natalie Wood played a six-year-old
in "Miracle on 34th Street,"
six factorial equals 720,
and Wood was born on 7/20, 1938.

"How I love music."

-- John O'Hara, Hope of Heaven, 1938

For related metaphors, see
Immortal Diamond,
The Diamond Archetype, and
the first log24.net entry...
for July 20, 2002.

Friday, January 2, 2004  2:14 PM

Dunne's Wake:

What, and Give Up Show Biz?

"Dying is easy. Comedy is hard."

-- Saying attributed to Edmund Gwenn, star of "Miracle on 34th Street," and also attributed to "Noel Coward, David Garrick, William Holden, Edmund Kean, Marcel Marceau, Groucho Marx, and Oscar Wilde."

See also yesterday's entry on the Dark Lady.  For more on Santa and the Dark Lady, see my archive for Aug.-Sept. 2002.

"Drink up, sweet.  You gotta go some.  How I love music.  Frère Jacques, Cuernavaca, ach du lieber August.  All languages.  A walking Berlitz.  Berlitz sounds like you with that champagne, my sweet, or how you're gonna sound."

-- Hope of Heaven, by John O'Hara,
"another acidic writer to whom he
[John Gregory Dunne]
was often compared"
(Adam Bernstein, Washington Post)

For some context for the Hope of Heaven quotation, see Immortal Diamond: O'Hara, Hopkins, and Joyce, or click on the adding machine in yesterday's entry.

For more on miracles and the afterlife, see my archive for September 2002.

Thursday, January 1, 2004  3:36 PM

The Dark Lady

"... though she has been seen by many men, she is known to only a handful of them.  You'll see her -- if you see her at all -- just after you've taken your last breath.  Then, before you exhale for the final time, she'll appear, silent and sad-eyed, and beckon to you.

She is the Dark Lady, and this is her story."

-- Mike Resnick

"... she played (very effectively) the Deborah Kerr part in a six-hour miniseries of From Here to Eternity...."

-- John Gregory Dunne on Natalie Wood
in the New York Review of Books
dated Jan. 15, 2004

Very effectively.