Monday, July 6, 2009
Art and Faith
Virginia Woolf, The Waves, Harvest Books paperback, 1950, pp. 248-249:
“On the outskirts of every agony sits some observant fellow who points; who whispers as he whispered to me that summer morning in the house where the corn comes up to the window, ‘The willow grows on the turf by the river. The gardeners sweep with great brooms and the lady sits writing.’ Thus he directed me to that which is beyond and outside our own predicament; to that which is symbolic, and thus perhaps permanent, if there is any permanence in our sleeping, eating, breathing, so animal, so spiritual and tumultuous lives.”
Up to the first semicolon, this is the Associated Press thought for today.
Related aesthetic philosophy from The Washington Post:
“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself, a church of American pragmatism that deals with the material stuff of experience in the history of art. To understand these lectures, which began promising an argument about how abstraction works and ended with an almost medieval allegory of how man confronts the void, one has to understand that Varnedoe views the history of abstraction as a pastor surveys the flock.”
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Friday, July 3, 2009
Damnation Morning
continued
“The tigers of wrath are wiser
than the horses of instruction.”
— Blake
“… the moment is not
properly an atom of time
but an atom of eternity.
It is the first reflection
of eternity in time, its first
attempt, as it were, at
stopping time….”
— Kierkegaard
Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
— Rubén Darío
Related material:
The deaths of
Ernest Hemingway
on the morning of
Sunday, July 2, 1961,
and of Alexis Arguello
on the morning of
Wednesday, July 1, 2009.
See also philosophy professor
Clancy Martin in the
London Review of Books
(issue dated July 9, 2009)
on AA members as losers—
“the ‘last men,’ the nihilists,
the hopeless ones.”
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Thursday, July 2, 2009
Hieron Grammaton, Part III*
The Old Man and the Light
In memory of
Ernest Hemingway,
who died on this date
in 1961, a story
in three parts:
I — Eye of Raven
Fermata
II — Psyche and Symbol
Leonard Baskin, detail of
cover for Jung’s
Psyche and Symbol
III — Raven Steals the Light
Detail from the story
“Raven Steals the Light”
Midrash:
“To the earnestness of death belongs precisely that capacity for awakening, that resonance of a profound mockery which, detached from the thought of the eternal, is an empty and often brash jest, but together with the thought of the eternal is just what it should be….” —Kierkegaard
* For Hieron Grammaton, Parts I and II, see the five Log24 entries from 6:29 PM Tuesday, June 23, to 1:00 AM Sunday, June 28.
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Wednesday, April 8, 2009
“Since 1963, when Pynchon’s first novel, V., came out, the writer– widely considered America’s most important novelist since World War II– has become an almost mythical figure, a kind of cross between the Nutty Professor (Jerry Lewis’s) and Caine in Kung Fu.”
— Nancy Jo Sales in the November 11, 1996, issue of New York Magazine
A Cross Between
(Click on images for their
source in past entries.)
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Monday, March 16, 2009
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Friday, February 27, 2009
Time and Chance
continued
Today's Pennsylvania lottery numbers suggest the following meditations…
Midday: Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."*
Evening: 5/07, 2008, in this journal– "Forms of the Rock."
Related material:
A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.
The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."
A related quotation by Lefebvre (cf. 2/22 2009):
"… an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."
— Page 25 of The Production of Space (Blackwell Publishing, 1991)
This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.
In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†
* Ending on Stephen King's birthday, 2008
† Mardi Gras
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Monday, February 23, 2009
Project MUSE —
… and interpretations, “any of the
Zingari shoolerim [gypsy schoolchildren]
may pick a peck of kindlings yet from the
sack of auld hensyne” (FW 112.4-8). …
— Patrick McGee, “Reading Authority:
Feminism and Joyce,” MFS: Modern
Fiction Studies— Volume 35, Number 3,
Fall 1989, pp. 421-436, The Johns Hopkins
University Press
|
McGee Thanks the Academy:
“The ulterior motive behind this essay [“Reading Authority,” above], the purpose for which I seize this occasion, concerns the question or problem of authority. I stress at the outset my understanding of authority as the constructed repository of value or foundation of a system of values, the final effect of fetishism– in this case, literary fetishism. [Cf. Marx, Das Kapital] Reading– as in the phrase ‘reading authority’– should be grasped as the institutionally determined act of constructing authority….”
Wikipedia:
“[In Peter Pan] Smee is Captain Hook’s right-hand man… Barrie describes him as ‘Irish’ and ‘a man who stabbed without offence‘….”
Background: In yesterday’s morning entry, James Joyce as Jesuit, with “Dagger Definitions.”
A different Smee appears as an art critic in yesterday’s afternoon entry “Design Theory.”–
Smee Stabs Without Offence:
“Brock, who has a brisk mind, is a man on a mission. He read mathematical economics and political philosophy at Princeton (he has five degrees in all) and is the founder and president of Strategic Economic Decisions Inc., a think tank specializing in applying the economics of uncertainty to forecasting and risk assessment.
But phooey to all that; Brock has deeper things to think about. He believes he has cracked the secret of beautiful design. He even has equations and graphs to prove it.”
A Jesuit in Portrait of the Artist as a Young Man:
“When may we expect to have something from you on the esthetic question?”
Beckett Bethicketted:
“Our entanglement in the wilderness of Finnegans Wake is exemplified by the neologism ‘Bethicket.’ This word condenses a range of possible meanings and reinforces a diversity of possible syntactic interpretations. Joyce seems to allude to Beckett, creating a portmanteau word that melds ‘Beckett’ with ‘thicket’ (continuing the undergrowth metaphor), ‘thick’ (adding mental density to floral density)…. As a single word ‘Bethicket’ contains the confusion that its context suggests. On the one hand, ‘Bethicket me for a stump of a beech’ has the sound of a proverbial expletive that might mean something like ‘I’ll be damned’ or ‘Well, I’ll be a son of a gun.’….”
— Stephen Dilks
At the Oscars, 2009
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Saturday, February 14, 2009
The Devil
in the Details
The Bloomsbury Auctions caption for these images is as follows:
303. Einstein (Albert, theoretical physicist, 1879-1955) Autograph Letter signed to Eric B. Gutkind, in German, 1½pp. & envelope, 4to, Princeton, 3rd January 1954, thanking him for a copy of his book and expressing his view of God and Judaism, [The word God is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honorable, but still primitive legends which are nevertheless pretty childish… . For me the Jewish religion like all other religions is an incarnation of the most childish superstitions. And the Jewish people to whom I gladly belong and with whose mentality I have a deep affinity have no different quality for me than all other people…], folds, slightly browned ; and a photograph of Gutkind, v.s., v.d.
est. £6000 – £8000
Einstein’s view of God and Judaism.
Eric B. Gutkind (1877-1965), philosopher; author of Choose Life: The Biblical Call to Revolt, 1952.
Albert Einstein – see also lot 497
Sold for £170000
Sale 649, 15th May 2008
Here is a close reading of the part of the letter itself that Bloomsbury gives in English, transcribed from the above images.
Line-by-line transcription of paragraph 2, starting at line 4 of that paragraph:
... Das Wort Gott ist für mich nichts als Ausdruck
und Produkt menschlicher Schwächen, die Bibel eine Sammlung
ehrwürdiger, aber doch reichlich primitiver Legenden. Keine noch
so feinsinnige Auslegung kann (für mich) etwas daran ändern.
Diese verfeinerten Auslegungen sind naturgemäß höchst mannigfaltig
und haben so gut wie nichts mit dem Urtext zu schaffen. Für
mich ist die unverfälschte jüdische Religion, wie alle anderen
Religionen, eine Inkarnation des primitiven Aberglaubens. Und das
jüdische Volk, zu dem ich gern gehöre und mit dessen Mentalität ich
tief verwachsen bin, hat für mich doch keine andersartige
Qualität als alle anderen Völker. So weit meine Erfahrung reicht,
ist es auch um nichts besser als andere menschliche Gruppierungen,
wenn es auch durch Mangel an Macht gegen die schlimmsten
Auswüchse gesichert ist. Ansonsten kann ich nichts "Auserwähltes"
an ihm wahrnehmen.
The Guardian of May 13, 2008 stated that the following was "translated from German by Joan Stambaugh"–
... The word God is for me nothing more than the expression
and product of human weaknesses, the Bible a collection
of honourable, but still primitive legends which are nevertheless pretty childish. No
interpretation no matter how subtle can (for me) change this.
These subtilised interpretations are highly manifold
according to their nature and have almost nothing to do with the original text. For
me the Jewish religion like all other
religions is an incarnation of the most childish [German: primitiven] superstitions. And the
Jewish people to whom I gladly belong and with whose mentality I
have a deep affinity have no different
quality for me than all other people. As far as my experience goes,
they are also no better than other human groups,
although they are protected from the worst
cancers by a lack of power. Otherwise I cannot see anything 'chosen'
about them.
Phrases by Stambaugh that do not appear in the German text are highlighted.
Stambaugh, a philosophy professor, is the author of a work on Buddhism, The Formless Self. For some related material on young men who "go crying 'The world is myself, life is myself'" in May, see Wallace Stevens's "The Pediment of Appearance."
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Monday, December 8, 2008
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Saturday, November 8, 2008
From a
Cartoon Graveyard
“That corpse you planted
last year in your garden,
Has it begun to sprout?
Will it bloom this year?
Or has the sudden frost
disturbed its bed?”
— T. S. Eliot, “The Waste Land“
Wikipedia:
“In the Roman Catholic tradition, the term ‘Body of Christ’ refers not only to the body of Christ in the spiritual realm, but also to two distinct though related things: the Church and the reality of the transubstantiated bread of the Eucharist….
According to the Catechism of the Catholic Church, ‘the comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body….’
….To distinguish the Body of Christ in this sense from his physical body, the term ‘Mystical Body of Christ’ is often used. This term was used as the first words, and so as the title, of the encyclical Mystici Corporis Christi of Pope Pius XII.”
Pope Pius XII:
“83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.”
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Monday, October 20, 2008
Me and My Shadow
Thoughts suggested by Saturday's entry–
"… with primitives the beginnings of art, science, and religion coalesce in the undifferentiated chaos of the magical mentality…."
— Carl G. Jung, "On the Relation of Analytical Psychology to Poetry," Collected Works, Vol. 15, The Spirit in Man, Art, and Literature, Princeton University Press, 1966, excerpted in Twentieth Century Theories of Art, edited by James M. Thompson.
For a video of such undifferentiated chaos, see the Four Tops' "Loco in Acapulco."
"Yes, you'll be goin' loco
down in Acapulco,
the magic down there
is so strong."
This song is from the 1988 film "Buster."
(For a related religious use of that name– "Look, Buster, do you want to live?"– see Fritz Leiber's "Damnation Morning," quoted here on Sept. 28.)
Art, science, and religion are not apparent within the undifferentiated chaos of the Four Tops' Acapulco video, which appears to incorporate time travel in its cross-cutting of scenes that seem to be from the Mexican revolution with contemporary pool-party scenes. Art, science, and religion do, however, appear within my own memories of Acapulco. While staying at a small thatched-roof hostel on a beach at Acapulco in the early 1960's, I read a paperback edition of Three Philosophical Poets, a book by George Santayana on Lucretius, Dante, and Goethe. Here we may regard art as represented by Goethe, science by Lucretius, and religion by Dante. For a more recent and personal combination of these topics, see Juneteenth through Midsummer Night, 2007, which also has references to the "primitives" and "magical mentality" discussed by Jung.
"The major structures of the psyche for Jung include the ego, which is comprised of the persona and the shadow. The persona is the 'mask' which the person presents [to] the world, while the shadow holds the parts of the self which the person feels ashamed and guilty about."
— Brent Dean Robbins, Jung page at Mythos & Logos
As for shame and guilt, see Malcolm Lowry's classic Under the Volcano, a novel dealing not with Acapulco but with a part of Mexico where in my youth I spent much more time– Cuernavaca.
Lest Lowry's reflections prove too depressing, I recommend as background music the jazz piano of the late Dave McKenna… in particular, "Me and My Shadow."
McKenna died on Saturday, the date of the entry that included "Loco in Acapulco." Saturday was also the Feast of Saint Luke.
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Saturday, July 19, 2008
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
"
A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Hermann Weyl on the hard core of objectivity:
"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)
Steven H. Cullinane on the symmetries of a 4×4 array of points:
|
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
Let us apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms?
|
The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
called a "colorful tale"–
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Tuesday, July 8, 2008
Translation
Yesterday's entry discussed T.E. Hulme— a co-founder, with Ezra Pound, of the Imagist school of poetry. Recent entries on randomness, using the New York Lottery as a source of examples, together with Hulme's approach to poetry discussed yesterday, suggest the following meditation– what Charles Cameron might call a "bead game."
Part I:
Ezra Pound on Imagism (from Gaudier-Brzeska,* 1916):
Three years ago in Paris I got out of a "metro" train at La Concorde, and saw suddenly a beautiful face, and then another and another, and then a beautiful child’s face, and then another beautiful woman, and I tried all that day to find words for what this had meant to me, and I could not find any words that seemed to me worthy, or as lovely as that sudden emotion. [….]
The "one image poem" is a form of super-position, that is to say, it is one idea set on top of another. I found it useful in getting out of the impasse in which I had been left by my metro emotion. I wrote a thirty-line poem, and destroyed it because it was what we call work "of second intensity." Six months later I made a poem half that length; a year later I made the following hokku-like sentence: —
"The apparition of these
faces in the crowd:
Petals, on a
wet, black bough."
I dare say it is meaningless unless one has drifted into a certain vein of thought. In a poem of this sort one is trying to record the precise instant† when a thing outward and objective transforms itself, or darts into a thing inward and subjective.
Part II:
Eleanor Goodman on translation (in a July 7, 2008, weblog entry, "Pound and Process: An Introduction"):
"… all translations exist on an axis. Indeed, they exist in a manifold of many axes intersecting. One axis is that of foreignness and familiarity. One axis is that of structural mimicry, another of melodic mimicry. And one axis is that of semantic fidelity."
Goodman's use of the word "manifold" here is of course poetic, not mathematical.
Part III:
New York Lottery, mid-day on July 7, 2008: 771.
Part IV:
A Google search on manifold 771 reveals that 771 is, according to Google's scanners, an alternate form (a "translation," via structural mimicry) of a script version of the letter M. (See Part V below.)
Part V:
Long version of a
one-image poem —
"Random apparition:
manifold translated."
This poem summarizes the
relationship (See Part IV above) of
the (apparently) random number 771
to the rather non-random concept of
a linear manifold:
[Such lines and planes have not
been, in mathematical language,
"translated."]
— Paul R. Halmos,
Finite Dimensional Vector Spaces,
Princeton University Press, 1948
Short version of the
above one-image poem —
771:
* Gaudier-Brzeska created the artifact shown on the cover of Solid Objects, a work of literary theory by Douglas Mao. For more on that artifact and on the New York Lottery, see Sermon for St. Peter's Day. "It is not in the premise that reality/ Is a solid…." –Wallace Stevens
† "I was like, Oh My God." —Poet Billy Collins at Chautauqua Institution, morning of July 7, 2008
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Sunday, June 29, 2008
Big Rock
"I'm going to hit this problem
with a big rock."
A professor discusses a poem by Wallace Stevens:
"Professor Eucalyptus in 'Ordinary Evening' XIV, for example, 'seeks/ God in the object itself,' but this quest culminates in his own choosing of 'the commodious adjective/ For what he sees… the description that makes it divinity, still speech… not grim/ Reality but reality grimly seen/ And spoken in paradisal parlance new'…."
"God in the object" seems
unlikely to be found in the
artifact pictured on the
cover of Mao's book:
I have more confidence
that God is to be found
in the Ping Pong balls of
the New York Lottery….
These objects may be
regarded as supplying
a parlance that is, if not
paradisal, at least
intelligible– if only in
the context of my own
personal experience.
|
June 28, 2008:
These numbers can, of course,
be interpreted as symbols of
the dates 6/29 and 5/30.
The last Log24 entry of
6/29 (St. Peter's Day):
"The rock cannot be broken.
It is the truth."
– Wallace Stevens,
"Credences of Summer"
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Sunday, May 25, 2008
Wechsler Cubes
"Confusion is nothing new."
— Song lyric, Cyndi Lauper
Part I:
Magister Ludi
Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–
"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."
Part II:
A Bulky Memorandum
From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–
A description of the work of Bill Wechsler, a fictional artist:
"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has
— End of page 168 —
opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.
The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."
Part III:
Wechsler Cubes
(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) –
Part IV:
A Magic Gallery
|
ZZ
WW
Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:
Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte
Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen
Numbers and Names,
Wording and Words
With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes
English translation
by Catherine Schelbert
A related poem:
Alphabets
by Hermann Hesse
From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone's command;
it is a game whose rules
are nice and neat.
But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens'
feet upon the snow.
He'd hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He'd see the way love burns
and anguish trembles,
He'd wonder, laugh,
shake with fear and weep
because beyond this cipher's
cross-barred keep
he'd see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
prisoner-fashion.
He'd think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He'll sigh, and smile,
and feel all right again.
— Hermann Hesse (1943),
"Buchstaben," from
Das Glasperlenspiel,
translated by
Richard and Clara Winston
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Comments Off on Sunday May 25, 2008
Sunday, May 18, 2008
From the Grave
DENNIS OVERBYE
in yesterday's New York Times:
"From the grave, Albert Einstein
poured gasoline on the culture wars
between science and religion this week…."
An announcement of a
colloquium at Princeton:
Above: a cartoon,
"Coxeter exhuming Geometry,"
with the latter's tombstone inscribed
"GEOMETRY
600 B.C. —
1900 A.D.
R.I.P."
The above is from
The Paradise of Childhood,
a work first published in 1869.
"I need a photo-opportunity,
I want a shot at redemption.
Don't want to end up a cartoon
In a cartoon graveyard."
— Paul Simon
Albert Einstein,
1879-1955:
"It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the 'merely-personal,' from an existence which is dominated by wishes, hopes and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation…."
— Autobiographical Notes, 1949
Related material:
A commentary on Tom Wolfe's
"Sorry, but Your Soul Just Died"–
"The Neural Buddhists," by David Brooks,
in the May 13 New York Times:
"The mind seems to have
the ability to transcend itself
and merge with a larger
presence that feels more real."
A New Yorker commentary on
a new translation of the Psalms:
"Suddenly, in a world without
Heaven, Hell, the soul, and
eternal salvation or redemption,
the theological stakes seem
more local and temporal:
'So teach us to number our days.'"
and a May 13 Log24 commentary
on Thomas Wolfe's
"Only the Dead Know Brooklyn"–
"… all good things — trout as well as
eternal salvation — come by grace
and grace comes by art
and art does not come easy."
— A River Runs Through It
"Art isn't easy."
— Stephen Sondheim,
quoted in
Solomon's Cube.
For further religious remarks,
consult Indiana Jones and the
Kingdom of the Crystal Skull
and The Librarian:
Return to King Solomon's Mines.
Comments Off on Sunday May 18, 2008
Friday, May 16, 2008
“From the grave, Albert Einstein poured gasoline on the culture wars between science and religion this week.
A letter the physicist wrote in 1954 to the philosopher Eric Gutkind, in which he described the Bible as ‘pretty childish’ and scoffed at the notion that the Jews could be a ‘chosen people,’ sold for $404,000 at an auction in London. That was 25 times the presale estimate.”
A less controversial Einstein-related remark:
“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”
— Hermann Weyl, “Relativity Theory as a Stimulus in Mathematical Research,” Proceedings of the American Philosophical Society, Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Comments Off on Friday May 16, 2008
Wednesday, April 23, 2008
Upscale Realism
or, "Have some more
wine and cheese, Barack."
(See April 15, 5:01 AM)
Allyn Jackson on Rebecca Goldstein
in the April 2006 AMS Notices (pdf)
|
"Rebecca Goldstein’s 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on everyday life. Her new book, Incompleteness: The Proof and Paradox of Kurt Gödel, is a volume in the 'Great Discoveries' series published by W. W. Norton….
In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work. The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped….
A member of the audience posed a question that has been on the minds of many of Goldstein’s readers: Is The Mind-Body Problem based on her own life? She did indeed study philosophy at Princeton, finishing her Ph.D. in 1976 with a thesis titled 'Reduction, Realism, and the Mind.' She said that while there are correlations between her life and the novel, the book is not autobiographical….
She… talked about the relationship between Gödel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, 'Einstein was a real mensch, and Gödel was very neurotic.' Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles– exiles from Europe and intellectual exiles. Gödel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as implying the sweeping view that 'everything is relative.' These misinterpretations irked both men, said Goldstein. 'Einstein and Gödel were realists and did not like it when their work was put to the opposite purpose.'"
|
Related material:
From Log24 on
March 22 (Tuesday of
Passion Week), 2005:
|
"'What is this Stone?' Chloe asked…. 'It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— Many Dimensions,
by Charles Williams, 1931
For more on this theme
appropriate to Passion Week —
Jews playing God — see
Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences
Research Institute
at the
Commonwealth Club,
San Francisco,
Tuesday, March 22.
Wine and cheese
reception at 5:15 PM
(San Francisco time).
|
From
UPSCALE,
a website of the
physics department at
the University of Toronto:
|
Mirror Symmetry
"The image [above]
is a depiction of
the universe as a
mirror image of God,
drawn by Robert Fludd
in the early 17th century.
The caption of the
upper triangle reads:
'That most divine and beautiful
counterpart visible below in the
flowing image of the universe.'
The caption of the
lower triangle is:
'A shadow, likeness, or
reflection of the insubstantial*
triangle visible in the image
of the universe.'"
* Sic. The original is incomprehensibilis, a technical theological term. See Dorothy Sayers on the Athanasian Creed and John 1:5.
|
For further iconology of the
above equilateral triangles,
see Star Wars (May 25, 2003),
Mani Padme (March 10, 2008),
Rite of Sping (March 14, 2008),
and
Art History: The Pope of Hope
(In honor of John Paul II
three days after his death
in April 2005).
Happy Shakespeare's Birthday.
Comments Off on Wednesday April 23, 2008
Wednesday, April 16, 2008
Poetry for Physicists:
The Gates of Hell
From the obituary of physicist John Archibald Wheeler at Princeton:
In the fall of 1967, he was invited to give a talk…. As he spoke, he… [mentioned] something strange… what he called a gravitationally completely collapsed object. But such a phrase was a mouthful, he said, wishing aloud for a better name. "How about black hole?" someone shouted from the audience.
That was it. "I had been searching for just the right term for months, mulling it over in bed, in the bathtub, in my car, wherever I had quiet moments," he later said. "Suddenly this name seemed exactly right." He kept using the term, in lectures and on papers, and it stuck.
From Log24 last year on this date ("Happy Birthday, Benedict XVI"):
"Know the one about the Demiurge and the Abridgment of Hope?"
— Robert Stone, A Flag for Sunrise, Knopf, 1981, the final page, 439
From Dante, The Inferno, inscription on the gates of Hell:
"Abandon all hope, ye who enter."
From Psychoshop, an unfinished novel by Alfred Bester completed by Roger Zelazny:
His manner was all charm and grace; pure cafe society….
He purred a chuckle. "My place. If you want to come, I'll show you."
"Love to. The Luogo Nero? The Black Place?"
"That's what the locals call it. It's really Buoco Nero, the Black Hole."
"Like the Black Hole of Calcutta?"
"No. Black Hole as in astronomy. Corpse of a dead star, but also channel between this universe and its next-door neighbor."
"Here? In Rome?"
"Sure. They drift around in space until they run out of gas and come to a stop. This number happened to park here."
"How long ago?"
"No one knows," he said. "It was there six centuries before Christ, when the Etruscans took over a small town called Roma and began turning it into the capital of the world."
Comments Off on Wednesday April 16, 2008
Saturday, February 23, 2008
Thanks to a Virginia reader for a reminder:
An Exercise
of Power
Johnny Cash:
"And behold,
a white horse."
Chess Knight
(in German, Springer)
"
Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"
Background of "Frau vBayern" from thePeerage.com:
Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg
F, #64640, b. 15 March 1978Last Edited=20 Oct 2005
The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:
From Log24 on
St. Bridget's Day, 2007:
The quotation
"Science is a Faustian bargain"
and the following figure–
Change
From a short story by
the above Princess:
"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."
To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in
"For Princeton's Class of 2007"—
Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–
Related material:
Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and
(Click on image for details.)
Comments Off on Saturday February 23, 2008
Friday, February 1, 2008
Kindergarten Theology
On the late
James Edwin Loder,
a Presbyterian minister and
a professor of Christian education
at Princeton Theological Seminary,
co-author of
The Knight’s Move (1992):
“At his memorial service his daughter Tami told the story of ‘little Jimmy,’ whose kindergarten teacher recognized a special quality of mind that set him apart. ‘Every day we read a story, and after the story is over, Jimmy gets up and wants to tell us what the story means.'” —
Dana R. WrightFor a related story about
knight moves and kindergarten,
see Knight Moves: The Relativity
Theory of Kindergarten Blocks,
and Log24, Jan. 16, 17, and 18.
See also Loder’s book
(poorly written, but of some
interest in light of the above):
Opening of
The Knight’s Move —
“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.
The significance of the title of this volume might stop there but for Kierkegaard’s use of the ‘knight’ image. The force of Kierkegaards’s usage might be described in relation to the chess metaphor by saying that not merely does Kierkegaard’s ‘knight of faith’ undertake a unique move within the rules of the human game, but faith transposes the whole idea of a ‘knight’s move’ into the mind of the Chess Master Himself. That is to say, chess is a game of multiple possibilities and interlocking strategies, so a chess master must combine the continuity represented by the whole complex of the game with the unpredictable decision he must make every time it is his turn. A master chess player, then, does not merely follow the rules; in him the game becomes a construct of consciousness. The better the player the more fully the game comes into its own as a creation of human intelligence. Similarly, for Kierkegaard, the knight of faith is a unique figure in human experience. The knight shows how, by existing in faith as a creative act of Christ’s Spirit, human existence comes into its own as an expression of the mind of Christ. Thus, the ultimate form of a ‘knight’s move’ is a creative act raised to the nth power by
Spiritus Creator, but it still partakes fully in the concrete pieces and patterns that comprise the nature of the human game and the game of nature.”
— James E. Loder and W. Jim Neidhardt (Helmers & Howard Publishing, 1992)
For a discussion, see Triplett’s
“Thinking Critically as a Christian.”
Many would deny that such
a thing is possible; let them
read the works of T. S. Eliot.
Related material:
The Knight’s Move
discusses (badly) Hofstadter’s
“strange loop” concept; see
Not Mathematics but Theology
(Log24, July 12, 2007).
Comments Off on Friday February 1, 2008
Thursday, January 3, 2008
Two reviews from the February 2008 Notices of the American Mathematical Society:
From a review of
A Certain Ambiguity
(A Mathematical Novel)
by Gaurav Suri and Hartosh Singh Bal
Princeton University Press
Hardcover, US$27.95, 281 pages —
“From the Habermas-Lyotard debate (see [1] for an introduction) to the Sokal hoax ([4]), to recent atheist manifestos on the bestseller lists (e.g., [2]) the question of foundations for intellectual thought and especially for intellectual debate has never been more critical or urgent.”
[1] M. Bérubé, What’s Liberal about the Liberal Arts? Classroom Politics and “Bias” in Higher Education, W. W. Norton, 2006.
[2] S. Harris, Letter to a Christian Nation, Knopf, 2006.
[4] A. Sokal and P. Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, 1999.
— Danny Calegari of Caltech
Also in the February Notices– a review of a book, Superior Beings: If They Exist, How Would We Know?, in which the author
“.. uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….
In the ‘Revelation Game,’ for example,
the Person (P) has two options:
1) P can believe in SB’s existence
2) P can not believe in SB’s existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself….
… [and] goals allow us to rank all the outcomes for each player from best… to worst…. The question we must answer is: what is the Nash equilibrium in this case?”
The answer is what one might expect from the American Mathematical Society:
“… the dominant strategy for both is when SB does not reveal Himself and P does not believe in His existence.”
Other strategies are, of course, possible. See last year’s entries.
See also
the life of John Nash,
for whom the above
equilibrium is named.
Comments Off on Thursday January 3, 2008
Sunday, December 30, 2007
The Christmas Tiger
Part I:
The Gauntlet
On Jonah Goldberg's new book Liberal Fascism– an attack on, among others, Woodrow Wilson:
"'… at some point,' Goldberg writes, 'it is necessary to throw down the gauntlet, to draw a line in the sand, to set a boundary, to cry at long last, "Enough is enough."'"
The Goldberg declaration is from a review in today's
New York Times titled "
Heil Woodrow!"
Part II:
Uncle Duke
Goes to Washington
Today's Doonesbury:
Part III:
A Holiday Tradition
Dialogue from the classic Capra film "Mr. Smith Goes to Washington"–
SUMMERS: When the country needs men up there who know and have courage as it never did before, he's just gonna decorate a chair and get himself honored.
DARRELL: Oh, but he'll vote! Sure. Just like his colleague tells him to.
DIZ: "Yes, sir," like a Christmas tiger. He'll nod his head and vote…
REPORTERS: "Yes."
DIZ: You're not a Senator! You're an honorary stooge! You ought to be shown up!
Part IV:
The Tigers of Princeton
The Christmas evening Pennsylvania Lottery 4-digit number was 0666, the Christian "number of the beast." For the beast itself, see the Dec. 3 Log24 entry "
Santa's Polar Opposite?" with its
link to a discussion of a metaphorical tiger at the South Pole.
A more realistic version of the beast appeared in the news on Christmas evening.
The Christmas number may also be interpreted as a reference to 6/6/6, the graduation date of the Class of 2006 at Princeton University.
Comments Off on Sunday December 30, 2007
Thursday, December 27, 2007
Chronicles
"Fullness… Multitude."
— The missing last words
of Inman in Cold Mountain,
added here on the
Feast of St. Luke, 2004
II Chronicles 1:
7: In that night did God appear unto Solomon, and said unto him, Ask what I shall give thee.
8: And Solomon said unto God, Thou hast shewed great mercy unto David my father, and hast made me to reign in his stead.
9: Now, O LORD God, let thy promise unto David my father be established: for thou hast made me king over a people like the dust of the earth in multitude.
10: Give me now wisdom and knowledge, that I may go out and come in before this people: for who can judge this thy people, that is so great?
On Kirk Varnedoe
"At 42– a professor with no museum experience– he was named curator of painting and sculpture at the Museum of Modern Art. It was, and is, the most influential job in the fluid, insular, fiercely contentious world of modern art. Just two decades past his last Amherst game, the lineman from Savannah was sitting in the chair where the most critical decisions in his profession are made– 'the conscientious, continuous, resolute distinction of quality from mediocrity,' according to his Olympian predecessor Alfred Barr. The Modern and its chief curator serve the American art establishment as a kind of aesthetic Supreme Court, and most of their rulings are beyond appeal."
— Hal Crowther
On Quality
Varnedoe, in his final
Mellon lecture at
the National Gallery,
quoted "Blade Runner"–
"I've seen things
you people wouldn't believe….
"Frank Rich of The New York Times
on the United States of America:
"A country where
entertainment is god."
Rich's description may or may not
be true of the United States, but
it certainly seems true of
The New York Times:
Click on image to enlarge.
Related material:
Art Wars
Comments Off on Thursday December 27, 2007
Monday, December 24, 2007
From Saturday's entry
(Log24, Dec. 22, 2007)
a link goes to–
The five entries of June 14, 2007.
From there, the link
"One Two Three Four,
Who Are We For?"
goes to–
Princeton: A Whirligig Tour
(Log24, June 5, 2007).
From there, the link
"Taking Christ to Studio 60"
goes to–
The five Log 24 entries
prior to midnight Sept. 18, 2006.
From there, the link
"Log24, January 18, 2004"
goes to–
A Living Church.
From there, the link
"click here"
goes to–
In the Bleak Midwinter
(Internet Movie Database)…
Tagline:
The drama. The passion. The intrigue… And the rehearsals haven't even started.
Plot Summary:
Out of work actor Joe volunteers to help try and save his sister's local church
for the community by putting on a Christmas production of Hamlet…
"… were it not that
I have bad dreams."
— Hamlet
Related material:
The New York Times online
obituaries of December 22,
Ike Turner's
"Bad Dreams" album
(see Log24, July 12, 2004),
"Devil Music," a composition
by H. S. M. Coxeter,
and
King of Infinite Space.
Those desiring more literary depth
may consult the G. K. Chesterton
play "Magic" for which Coxeter
wrote his "Devil Music" and
the Ingmar Bergman film
"The Magician" said to have
been inspired by Chesterton.
Comments Off on Monday December 24, 2007
Tuesday, October 16, 2007
In memory of
Harish-Chandra,
who died at 60
on this date in 1983
Harish-Chandra in 1981
(Photo by Herman Landshof)
Recent Log24 entries have parodied the use of the phrase “deep beauty” as the title of the
Oct. 3-4 physics symposium of that name, which was supported by a grant from the John Templeton Foundation and sponsored by the Department of Philosophy at Princeton University.
Such parody was in part suggested by the symposium’s sources of financial and academic support. This support had, in the view of some, the effect of linking the symposium’s topic, the mathematics of quantum theory, with both religion (
the Templeton Foundation) and philosophy (a field sometimes associated in popular thought– though
not at Princeton— with
quantum mysticism.)
As a corrective to the previous parodies here, the following material on the mathematician Harish-Chandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.
MacTutor History of Mathematics:
“Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.”
R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):
“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semi-simple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”
“It is difficult to communicate the grandeur of Harish-Chandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”
For deeper views of his work, see
- Rebecca A. Herb, “Harish-Chandra and His Work” (pdf), Bulletin of the American Mathematical Society, July 1991, and
- R. P. Langlands, “Harish-Chandra, 1923-1983” (pdf, 28 pp., Royal Society memoir, 1985)
Comments Off on Tuesday October 16, 2007
Monday, October 1, 2007
Bright as Magnesium
"Definitive"
— The New York Times,
Sept. 30, 2007, on
Blade Runner:
The Final Cut
Institute for Advanced Study, Princeton, N.J.—
"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."
Hal Crowther—
"His final lecture was an eloquent, prophetic flight of free association….
Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"
Related material:
tears in the rain–
Game Over
(Nov. 5, 2003):
Comments Off on Monday October 1, 2007
Saturday, August 11, 2007
Four Colours
The previous entry dealt with Plato’s myth of the ring of Gyges that conferred invisibility. Another legendary ring, from Hermann Hesse, with some background from Carl Jung:
From C. G. Jung, Collected Works (Princeton U. Press), Volume 12– Psychology and Alchemy (1944)– Part II– “Individual Dream Symbolism in Relation to Alchemy”– Chapter 3, “The Symbolism of the Mandala”– as quoted in Jung, Dreams, published by Routledge, 2001– Page 265–
“… the dreamer is wandering about in a dark cave, where a battle is going on between good and evil. But there is also a prince who knows everything. He gives the dreamer a ring set with a diamond….
Visual impression (waking dream):
The dreamer is falling into the abyss. At the bottom there is a bear whose eyes gleam alternately in four colours: red, yellow, green, and blue. Actually it has four eyes that change into four lights. The bear disappears and the dreamer goes through a long dark tunnel. Light is shimmering at the far end. A treasure is there, and on top of it the ring with the diamond. It is said that this ring will lead him on a long journey to the east.”
Hermann Hesse, The Journey to the East (1932):
“‘… Despair is the result of each earnest attempt to go through life with virtue, justice, and understanding and to fulfil their requirements. Children live on one side of despair, the awakened on the other side. Defendant H. is no longer a child and is not yet fully awakened. He is still in the midst of despair. He will overcome it and thereby go through his second novitiate. We welcome him anew into the League, the meaning of which he no longer claims to understand. We give back to him his lost ring, which the servant Leo has kept for him.’
The Speaker then brought the ring, kissed me on the cheek and placed the ring on my finger. Hardly had I looked at the ring, hardly had I felt its metallic coolness on my fingers, when a thousand things occurred to me, a thousand inconceivable acts of neglect. Above all, it occurred to me that the ring had four stones at equal distances apart, and that it was a rule of the League and part of the vow to turn the ring slowly on the finger at least once a day, and at each of the four stones to bring to mind one of the four basic precepts of the vow. I had not only lost the ring and had not once missed it, but during all those dreadful years I had also no longer repeated the four basic precepts or thought of them. Immediately, I tried to say them again inwardly. I had an idea what they were, they were still within me, they belonged to me as does a name which one will remember in a moment but at that particular moment cannot be recalled. No, it remained silent within me, I could not repeat the rules, I had forgotten the wording. I had forgotten the rules; for many years I had not repeated them, for many years I had not observed them and held them sacred– and yet I had considered myself a loyal League brother.
The Speaker patted my arm kindly when he observed my dismay and deep shame.”
Comments Off on Saturday August 11, 2007
Thursday, June 28, 2007
Real Numbers:
An Object Lesson
A Cornell professor discusses a poem by Wallace Stevens:
"Professor Eucalyptus in 'Ordinary Evening' XIV, for example, 'seeks/ God in the object itself,' but this quest culminates in his own choosing of 'the commodious adjective/ For what he sees… the description that makes it divinity, still speech… not grim/ Reality but reality grimly seen/ And spoken in paradisal parlance new'…."
"God in the object" seems
unlikely to be found in the
artifact pictured on the
cover of Mao's book:
I have more confidence
that God is to be found
in the Ping Pong balls
of the New York Lottery.
These objects may be
regarded as supplying
a parlance that is, if not
paradisal, at least
intelligible– if only in
the context of my own
personal experience:
Journal entry dated 5/14:
Comments Off on Thursday June 28, 2007
Thursday, June 7, 2007
Masters of Chaos
From the May 6, 2007,
New York Times,
Charles McGrath on
Philip K. Dick:
His early novels, written in two weeks or less, were published in double-decker Ace paperbacks that included two books in one, with a lurid cover for each. "If the Holy Bible was printed as an Ace Double," an editor once remarked, "it would be cut down to two 20,000-word halves with the Old Testament retitled as 'Master of Chaos' and the New Testament as 'The Thing With Three Souls.'"
Click to enlarge.
As for "the thing with
three souls"–
Part I:
"Educate, Empower, Entertain"
— Motto of Yolanda King
Part II:
Three universities
(but not those of
Martin Myerson)–
Princeton, Harvard, Cambridge
Comments Off on Thursday June 7, 2007
Tuesday, June 5, 2007
Devil in the Details
This morning was the
Princeton commencement.
Meanwhile…
Comments Off on Tuesday June 5, 2007
Princeton:
A Whirligig Tour
Symbol from a
website on
“Presbyterian
Creedal Standards”
The above symbol
appeared here
on 11/8/02.
Related material:
1. The remarks of
Bradley Whitford
at Princeton’s
Class Day yesterday:
2. An illustration from
Log 24 on 11/10/06:
3. The Whirligig of Time
(1/5/03):
4. Natalie Angier, priestess of Scientism
(5/26/07), and her new book
The Canon: A Whirligig Tour of
the Beautiful Basics of Science
(available as a special from
Amazon.com):
Comments Off on Tuesday June 5, 2007
Friday, April 27, 2007
It’s still the
same old story…
From today’s online
New York Times:
Photo by Carol T. Powers
for The New York Times
Also in today’s online Times:
“Mstislav Rostropovich, a cellist and conductor who was renowned not only as one of the great instrumentalists of the 20th century, but also as an outspoken champion of artistic freedom in Russia during the final decades of the Cold War, died in Moscow today. He was 80 and lived in Paris, with homes in Moscow, St. Petersburg, London and Lausanne, Switzerland….
Mr. Rostropovich… was widely known by his diminutive, Slava (which means glory in Russian)….”
Related material:
I. “Established on 8 November 1943, the Order of Glory (Orden Slavy – Орден Славы) was an Order (decoration) of the Soviet Union…. The Order of Glory… was modelled closely upon the Tsarist Cross of St. George….” —Wikipedia
II. Also on the 8th of November, in 2006 and 2002: Grave Matters and Religious Symbolism at Princeton.
III. “Mr. Rostropovich will be buried in Moscow at the Novodevichy Cemetery, where on Wednesday his friend, Boris Yeltsin, post-Soviet Russia’s first president, was laid to rest.” —New York Times
IV. “A graveyard smash.” –Bobby (Boris) Pickett, who died Wednesday.
Comments Off on Friday April 27, 2007
Tuesday, April 3, 2007
Our Judeo-Christian
Heritage –
Lottery
Hermeneutics
Part II: Christian
Part III:
Imago Dei
Click on picture
for details.
Related material:
It is perhaps relevant to
this Holy Week that the
date 6/04 (2006) above
refers to both the Christian
holy day of Pentecost and
to the day of the
facetious baccalaureate
of the Class of 2006 in
the University Chapel
at Princeton.
For further context for the
Log24 remarks of that same
date, see June 1-15, 2006.
Comments Off on Tuesday April 3, 2007
Wednesday, March 21, 2007
Finite Relativity
continued
This afternoon I added a paragraph to
The Geometry of Logic that makes it, in a way, a sequel to the webpage
Finite Relativity:
"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."
The epigraph to "Finite Relativity" is:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
The added paragraph seems to fit this description.
Comments Off on Wednesday March 21, 2007
Tuesday, February 20, 2007
Symmetry
Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”
Some relevant quotations:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Describing the branch of mathematics known as Galois theory, Weyl says that it
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
Weyl’s set Sigma is a finite set of complex numbers. Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations. For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:
“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].‘[22]
22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).
References:
Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.
Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]
Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.
See also
Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–
“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–
“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.
Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].
Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick.
Comments Off on Tuesday February 20, 2007
Friday, February 16, 2007
“The much-borrowed Brown formula involves some very specific things. The name of a great artist, artifact or historical figure must be in the book’s story, not to mention on its cover. The narrative must start in the present day with a bizarre killing, then use that killing as a reason to investigate the past. And the past must yield a secret so big, so stunning, so saber-rattling that all of civilization may be changed by it. Probably not for the better.
This formula is neatly summarized….”
Cover illustration
for
The Judas Seat:
The Narrative:
The Secret:
Part I
“Little ‘Jack’ Horner was actually Thomas Horner, steward to the Abbot of Glastonbury during the reign of King Henry VIII…. Always keen to raise fresh funds, Henry had shown a interest in Glastonbury (and other abbeys). Hoping to appease the royal appetite, the nervous Abbot, Richard Whiting, allegedly sent Thomas Horner to the King with a special gift. This was a pie containing the title deeds to twelve manor houses in the hope that these would deflect the King from acquiring Glastonbury Abbey. On his way to London, the not so loyal courier Horner apparently stuck his thumb into the pie and extracted the deeds for Mells Manor, a plum piece of real estate. The attempted bribe failed and the dissolution of the monasteries (including Glastonbury) went ahead from 1536 to 1540. Richard Whiting was subsequently executed, but the Horner family kept the house, so the moral of this one is: treachery and greed pay off, but bribery is a bad idea.” –Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind the Rhyme
“In medieval romance, the grail was said to have been brought to
Glastonbury in Britain by Joseph of Arimathea and his followers. In the time of Arthur, the quest for the Grail was the highest spiritual pursuit.” —
The Camelot Project
Part III
The Log24 entry for the date–
February 13, 2007–
of the above Bible scholar’s death,
and the three entries preceding it:
“And what the dead had no speech for, when living,
they can tell you, being dead:
the communication of the dead is tongued with fire
beyond the language of the living.”
— T. S. Eliot, Four Quartets
Comments Off on Friday February 16, 2007
Friday, February 2, 2007
The Night Watch
For Catholic Schools Week
(continued from last year)–
Last night’s Log24 Xanga
footprints from Poland:
Poland 2/2/07 1:29 AM
/446066083/item.html
2/20/06: The Past Revisited
(with link to online text of
Many Dimensions, by Charles Williams)
Poland 2/2/07 2:38 AM
/426273644/item.html
1/15/06 Inscape
(the mathematical concept, with
square and “star” diagrams)
Poland 2/2/07 3:30 AM
nextdate=2%252f8%252f20…
2/8/05 The Equation
(Russell Crowe as John Nash
with “star” diagram from a
Princeton lecture by Langlands)
Poland 2/2/07 4:31 AM
/524081776/item.html
8/29/06 Hollywood Birthday
(with link to online text of
Plato on the Human Paradox,
by a Fordham Jesuit)
Poland 2/2/07 4:43 AM
/524459252/item.html
8/30/06 Seven
(Harvard, the etymology of the
word “experience,” and the
Catholic funeral of a professor’s
23-year-old daughter)
Poland 2/2/07 4:56 AM
/409355167/item.html
12/19/05 Quarter to Three (cont.)
(remarks on permutation groups
for the birthday of Helmut Wielandt)
Poland 2/2/07 5:03 AM
/490604390/item.html
5/29/06 For JFK’s Birthday
(The Call Girls revisited)
Poland 2/2/07 5:32 AM
/522299668/item.html
8/24/06 Beginnings
(Nasar in The New Yorker and
T. S. Eliot in Log24, both on the 2006
Beijing String Theory conference)
Poland 2/2/07 5:46 AM
/447354678/item.html
2/22/06 In the Details
(Harvard’s president resigns,
with accompanying “rosebud”)
Comments Off on Friday February 2, 2007
Saturday, January 6, 2007
Picture of Nothing
On Kirk Varnedoe’s
2003 Mellon Lectures,
“Pictures of Nothing“–
“Varnedoe’s lectures were ultimately
about faith, about his faith in
the power of abstraction,
and abstraction as a kind of
anti-religious faith in itself….”
— The Washington Post
Related material:
The more industrious scholars
will derive considerable pleasure
from describing how the art-history
professors and journalists of the period
1945-75, along with so many students,
intellectuals, and art tourists of every
sort, actually struggled to see the
paintings directly, in the old
pre-World War II way,
like Plato’s cave dwellers
watching the shadows, without
knowing what had projected them,
which was the Word.”
— Tom Wolfe, The Painted Word
Log24, Aug. 23, 2005:
“Concept (scholastics’ verbum mentis)–
theological analogy of Son’s procession
as Verbum Patris, 111-12″
— Index to Joyce and Aquinas,
by William T. Noon, S.J.,
Yale University Press 1957,
second printing 1963, page 162
“So did God cause the big bang?
Overcome by metaphysical lassitude,
I finally reach over to my bookshelf
for The Devil’s Bible.
Turning to Genesis I read:
‘In the beginning
there was nothing.
And God said,
‘Let there be light!’
And there was still nothing,
but now you could see it.'”
— Jim Holt, Big-Bang Theology,
Slate‘s “High Concept” department
“Bang.”
“…Mondrian and Malevich
are not discussing canvas
or pigment or graphite or
any other form of matter.
They are talking about
Being or Mind or Spirit.
From their point of view,
the grid is a staircase
to the Universal….”
— Rosalind Krauss, “Grids”
For properties of the
“nothing” represented
by the 3×3 grid, see
The Field of Reason.
For religious material related
to the above and to Epiphany,
a holy day observed by some,
see Plato, Pegasus, and the
Evening Star and Shining Forth.
Comments Off on Saturday January 6, 2007
Thursday, November 16, 2006
“Let all thy words be counted.”
Dante, Inf., canto X.
Words for
G. Robert Crowningshield,
a developer of the International Diamond
Grading System™
According to a
press release,
Crowningshield
died on
November 8.
See Grave Matters,
an entry of that date,
and its links to
Geometry’s Tombstones,
Birth, Death, and Symmetry,
and
Religious Symbolism
at Princeton.
| Dante, Inferno, Canto X, 37-39:
E l’animose man del duca e pronte
mi spinser tra le sepulture a lui,
dicendo: “Le parole tue sien conte.”
And the bold and ready hands
of my Leader
pushed me between the tombs to him,
saying: “Let thy words be fitting”.
“Make your words count,”
Virgil instructs Dante:
“Speak aptly, make what you say
appropriate to the situation.”
|
Perhaps Crowningshield’s
Leader will be…
Niemoller is noted for his role in
the movement that led to the
Barmen Declaration, discussed in
Presbyterian Creedal Standards—
linked to in the above-cited
Religious Symbolism
at Princeton
(…that lay in the house
that Jack built).
Comments Off on Thursday November 16, 2006
Wednesday, November 1, 2006
Clifford Geertz
Professor Emeritus,
Institute for Advanced Study
Savage Logic
"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought. Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'
These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore. (How, precisely, they have come into being in the first place is one of the points on which Levi-Strauss is not too explicit, referring to them vaguely as the 'residue of events… fossil remains of the history of an individual or a society.') Such images are inevitably embodied in larger structures– in myths, ceremonies, folk taxonomies, and so on– for, as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular. But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort. Quoting Franz Boas that 'it would seem that mythological worlds have been built up, only to be shattered again, and that new worlds were built from the fragments,' Levi-Strauss generalizes this permutational view of thinking to savage thought in general."
— Clifford Geertz, "The Cerebral Savage: the Structural Anthropology of Claude Levi-Strauss," in Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.
Today's New York Times
reports that
Geertz died on Monday,
October 30, 2006.
Related material:
Kaleidoscope Puzzle,
Being Pascal Sauvage,
and Up the River:
|
While it's a story that's never been written, a suggested title– Indiana Jones Sails Up The River Of Death–
shows how readily we as individuals or we as a culture can automatically visualize a basic story motif. We may each see the particular elements of the story differently, but almost instantaneously we catch its drift.
The hero sails up the river of death to discover what lies within his own heart: i.e., how much moral and physical strength he has.
Indiana Jones sails up the River of Death.
We are following Indiana Jones up the River of Death. We're going to visit with Colonel Kurtz. (You may not want to get off the boat.)
No, I am not mixing up metaphors.
These are the Story.
|
Amen.
Comments Off on Wednesday November 1, 2006
Sunday, October 22, 2006
Go Tigers!
On this date:
“In 1746,
Princeton University
in New Jersey received
its charter.”
— Today in History
by The Associated Press
“The charter… authorized
the erection of a college….”
— Princeton University
Comments Off on Sunday October 22, 2006
Sunday, September 17, 2006
At Midnight
“At midnight
on the Emperor’s pavement flit
Flames that no faggot feeds,
nor steel has lit,
Nor storm disturbs, flames
begotten of flame,
Where blood-begotten spirits come
And all complexities of fury leave,
Dying into a dance,
An agony of trance,
An agony of flame that cannot
singe a sleeve.”
— From Byzantium, by
William Butler Yeats
“The only hope, or else despair
Lies in the choice of pyre or pyre–
To be redeemed from fire by fire.”
— From Four Quartets, by
Thomas Stearns Eliot
“Look around you. There is an eerie sense of Panic in the air, a silent Fear and Uncertainty that comes with once reliable faiths and truths and solid Institutions that are no longer safe to believe in…”
— Prepare for the Weirdness, by Hunter S. Thompson, quoted in a sermon for Pentecost Sunday, 2005
“If you passed, you got to live, and if you failed you were burned alive on a pyre that’s now the Transgender Studies Building.”
— Baccalaureate address at the interfaith worship service, Princeton University, on Pentecost Sunday, June 4, 2006
Review:
“At midnight on the Emperor’s pavement….”
Comments Off on Sunday September 17, 2006
Sunday, August 13, 2006
Comments Off on Sunday August 13, 2006
Tuesday, August 1, 2006
Highway 1Revisited
John Constantine,
cartoon character, and
Donald E. Knuth,
Lutheran mathematician
“I need a photo-opportunity,
I want a shot at redemption.
Don’t want to end up a cartoon
In a cartoon graveyard.”
Mel Gibson,
7/28/06,
photo by
Los Angeles County
Sheriff’s Department
This meditation is prompted by memories of suicidal alcoholics Hunter S. Thompson and Ernest Hemingway, as well as by the title of Mel Gibson’s latest project, “Apocalypto.”
A search on Gibson’s film title leads to this quotation:
“And what does apocalypse mean? It means revelation: apocalypto means to open up and to show the truth. But it also means absolute violence, so the apocalypse is a violent revelation and a revelation of violence and immediately you see the relevance of this.”
— Interview with Rene Girard in the June 1996 issue of UCLA’s Anthropoetics: The Journal of Generative Anthropology
It is by no means clear that “apocalypse” means “violence,” let alone “absolute violence,” except in the Christian tradition.
For apocalyptic Christian violence, see “Apocalypse and Violence: The Evidence from the Reception History of the Book of Revelation” (pdf), by Christopher Rowland of Oxford University.
As for “the relevance of this,” see the definition of “generative anthropology” (GA) at
anthropoetics.ucla.edu/purpose.htm:
“The originary hypothesis of GA is that human language begins as an aborted gesture of appropriation representing–and thereby renouncing as sacred– an object of potential mimetic rivalry. The strength of our mimetic intelligence makes us the only creatures for whom intraspecific violence is a greater threat to survival than the external forces of nature. Human language defers potential conflict by permitting each to possess the sign of the unpossessable object of desire– the deferral of violence through representation.”
Compare with the remarks of Jung on Transformation Symbolism in the Mass:
Antecedents and parallels are found for the ritual of the Christian religious Mass in Aztec, Mithraic and pagan religious practices. “The Aztecs make a dough figure of the god Huitzilopochtli, which is then symbolically killed, divided and consumed….”
Collected Works of C. G. Jung, Vol. 11. 2nd ed., Princeton University Press, 1969. (pp. 222-225)
Mel Gibson’s interest in religion and violence is well known. His film “Apocalypto,” scheduled for release on Dec. 8, 2006, deals with human sacrifice among the Maya, rather than the Aztecs or Jews. (Cf. Abraham and “Highway 61 Revisited.”)
It seems unlikely that Mel will learn more about these issues in his recovery program. Too bad.
Comments Off on Tuesday August 1, 2006
Tuesday, June 6, 2006
Pentecost
and
Queer Theory
“Stuff comes up,
weird doors open,
people fall into things.”
— David Sedaris,
baccalaureate address
at Princeton on Sunday,
June 4, 2006,
the Feast of Pentecost
“The truth is that man’s capacity for symbol-mongering in general and language in particular is so intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness itself, that it is all but impossible for him to focus on the magic prism through which he sees everything else.”
— Walker Percy, The Message in the Bottle: How Queer Man Is, How Queer Language Is, and What One Has to Do with the Other. New York: Farrar, Straus, and Girox, 1975, page 29.
Comments Off on Tuesday June 6, 2006
Monday, June 5, 2006
Sermon
Baccalaureate:
A farewell address
in the form of a sermon
delivered to a graduating class.
"Stuff comes up,
weird doors open,
people fall into things."
— David Sedaris,
baccalaureate address
at Princeton yesterday
"The truth is that man's capacity for symbol-mongering in general and language in particular is so intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness itself, that it is all but impossible for him to focus on the magic prism through which he sees everything else."
— Walker Percy, The Message in the Bottle: How Queer Man Is, How Queer Language Is, and What One Has to Do with the Other. New York: Farrar, Straus, and Giroux, 1975, page 29.
Review: ART WARS
on Sept. 12, 2002:
Und was für ein Bild des Christentums
ist dabei herausgekommen?
Voilà:
Related material:
Bright Star.
Comments Off on Monday June 5, 2006
Sunday, June 4, 2006
Images
and Words
for Baccalaureate
Day at Princeton
From Hermann Weyl’s
Symmetry,
Princeton University
Press, page 140
Adapted from the
cover of Alan Watts’s
The Spirit of Zen
Romani flag, courtesy of
myspace.com/RomArmando
Related material:
“The Scholar Gypsy”
in The Oxford Book
of English Prose, 1923,
edited by
Sir Arthur Quiller-Couch
This is available online:
From The Vanity of Dogmatizing,
by Joseph Glanvill
(London, printed by E.C. for
Henry Eversden at the Grey-Hound
in St.Pauls-Church-Yard, 1661)
Pages 195-201:
That one man should be able to bind the thoughts of another, and determine them to their particular objects; will be reckon’d in the first rank of Impossibles: Yet by the power of advanc’d Imagination it may very probably be effected; and story abounds with Instances. I’le trouble the Reader but with one; and the hands from which I had it, make me secure of the truth on’t. There was very lately a Lad in the University of Oxford, who being of very pregnant and ready parts, and yet wanting the encouragement of preferment; was by his poverty forc’d to leave his studies there, and to cast himself upon the wide world for a livelyhood. Now, his necessities growing dayly on him, and wanting the help of friends to relieve him; he was at last forced to joyn himself to a company of Vagabond Gypsies, whom occasionally he met with, and to follow their Trade for a maintenance. Among these extravagant people, and by the insinuating subtilty of his carriage, he quickly got so much of their love, and esteem; as that they discover’d to him their Mystery: in the practice of which, by the pregnancy of his wit and parts he soon grew so good a proficient, as to be able to out-do his Instructors. After he had been a pretty while exercis’d in the Trade; there chanc’d to ride by a couple of Scholars who had formerly bin of his acquaintance. The Scholars had quickly spyed out their old friend, among the Gypsies; and their amazement to see him among such society, had well-nigh discover’d him: but by a sign he prevented their owning him before that Crew: and taking one of them aside privately, desired him with his friend to go to an Inn, not far distant thence, promising there to come to them. They accordingly went thither, and he follows: after their first salutations, his friends enquire how he came to lead so odd a life as that was, and to joyn himself with such a cheating beggarly company. The Scholar-Gypsy having given them an account of the necessity, which drove him to that kind of life; told them, that the people he went with were not such Impostours as they were taken for, but that they had a traditional kind of learning among them, and could do wonders by the power of Imagination, and that himself had learnt much of their Art, and improved in further than themselves could. And to evince the truth of what he told them, he said, he’d remove into another room, leaving them to discourse together; and upon his return tell them the sum of what they had talked of: which accordingly he perform’d, giving them a full acount of what had pass’d between them in his absence. The Scholars being amaz’d at so unexpected a discovery, ernestly desir’d him to unriddle the mystery. In which he gave them satisfaction, by telling them, that what he did was by the power of Imagination, his Phancy binding theirs; and that himself had dictated to them the discourse, they held together, while he was from them: That there were warrantable wayes of heightening the Imagination to that pitch, as to bind anothers; and that when he had compass’d the whole secret, some parts of which he said he was yet ignorant of, he intended to give the world an account of what he had learned.
Now that this strange power of the Imagination is no Impossibility; the wonderful signatures in the Foetus caus’d by the Imagination of the Mother, is no contemptible Item. The sympathies of laughing & gaping together, are resolv’d into this Principle: and I see not why the phancy of one man may not determine the cogitation of another rightly qualified, as easily as his bodily motion. This influence seems to be no more unreasonable, then [sic] that of one string of a Lute upon another; when a stroak on it causeth a proportionable motion in the sympathizing confort, which is distant from it and not sensibly touched. Now if this notion be strictly verifiable; ’twill yeeld us a good account of how Angels inject thoughts into our minds, and know our cogitations: and here we may see the source of some kinds of fascination. If we are prejudic’d against the speculation, because we cannot conceive the manner of so strange an operation; we shall indeed receive no help from the common Philosophy: But yet the Hypothesis of a Mundane soul, lately reviv’d by that incomparable Platonist and Cartesian, Dr. H. More, will handsomely relieve us. Or if any would rather have a Mechanical account; I think it may probably be made out some such way as follow. Imagination is inward Sense. To Sense is required a motion of certain Filaments of the Brain; and consequently in Imagination there’s the like: they only differing in this, that the motion of the one proceeds immediately from external objects; but that of the other hath its immediate rise within us. Now then, when any part of the Brain is stringly agitated; that, which is next and most capable to receive the motive Impress, must in like manner be moved. Now we cannot conceive any thing more capable of motion, then the fluid matter, that’s interspers’d among all bodies, and contiguous to them. So then, the agitated parts of the Brain begetting a motion in the proxime Aether; it is propagated through the liquid medium, as we see the motion is which is caus’d by a stone thrown into the water. Now, when the thus moved matter meets with anything like that, from which it received its primary impress; it will proportionably move it, as it is in Musical strings tuned Unisons. And thus the motion being convey’d, from the Brain of one man to the Phancy of another; it is there receiv’d from the instrument of conveyance, the subtil matter; and the same kind of strings being moved, and much of whay after the same manner as in the first Imaginant; the Soul is awaken’d to the same apprehensions, as were they that caus’d them. I pretend not to any exactness or infallibility in this account, fore-seeing many scruples that must be removed to make it perfect: ‘Tis only a hint of the possibility of mechanically solving the Phaenomenon; though very likely it may require many other circumstances completely to make it out. But ’tis not my business here to follow it: I leave it therefore to receive accomplishment from maturer Inventions.
Comments Off on Sunday June 4, 2006
Friday, May 19, 2006
Women-Only Meeting at Princeton
From May 15 through May 26, there is a women-only meeting on zeta functions at the Institute for Advanced Study in Princeton, New Jersey. Today’s activities:
| 8:00- 9:45 a.m. |
Breakfast (Dining Hall) |
| 9:00- 9:30 a.m. |
T-shirt Sale, Harry’s Bar – Dining Hall |
| 9:30-10:00 a.m. |
Depart for Princeton University (talks, lunch, campus and art museum tour, and dinner) |
No movie?
Comments Off on Friday May 19, 2006
Thursday, May 18, 2006
Space Cadet
“They should have
sent a poet.”
You’d never know it,
But buddy, I’m a kind of poet…
Starring in tonight’s
New York Times obits:
|
One for
my baby
|
And one more
for the road
|
The
“Coppertone girl” artist died at 88 on Monday, May 15. In her memory, here is a
an entry from that day on
a women-only mathematics program at Princeton that started on Monday. That entry opens with a quote from Robert A. Heinlein, whose
writing inspired the TV series “Space Cadet.” The star of that series,
Frankie Thomas, died at 85 on Thursday, May 11. In his honor as a member of the elite Solar Guard, here is
a solar entry from Sunday, May 14. In honor of Jodie Foster, space explorer and Coppertone girl incarnate, here is a link to “
Spare Oom.”
“My God, it’s
full of stars!”
Comments Off on Thursday May 18, 2006
Monday, April 10, 2006
"What other colleges call fraternities,
Princeton calls Eating Clubs."
Illustrated below:
The Restaurant Quarré in Berlin,
with a view of the Brandenburg Gate.
Related etymology:
OF. quarré square, F. carré,
from L. quadratus square…
— Webster's Revised
Unabridged Dictionary, 1913
Related material:
(1) A symbol of symmetry
that might have pleased
Hermann Weyl:
Source —
Timothy A. Smith on
Bach's Fugue No. 21,
the Well-Tempered
Clavier, Book II
(pdf or Shockwave)
(2) The remarks of Noam D. Elkies
on his
"Brandenburg Concerto No. 7":
"It is of course an act of chutzpah,
some would say almost heresy,
to challenge Bach so explicitly
on his own turf."
(3) The five Log24 entries
culminating on Pi Day,
March 14, 2006
(4) The following event at the
Harvard University
mathematics department
on March 14, 2006, also
featuring Noam D. Elkies:
"At 3:14 p.m., six contestants began
a pie-eating contest…. Contestants had
exactly three minutes and 14 seconds
to eat as much pie as they could.
'Five, four, pi, three, two, one,'
Elkies counted down as the
contestants shoved the last
mouthful of pie
into their mouths…."
Noam D. Elkies
(5) The Magic Schmuck
Comments Off on Monday April 10, 2006
Sunday, March 26, 2006
'Nauts
(continued from
Life of the Party, March 24)
Exhibit A —
From (presumably) a Princeton student
(see Activity, March 24):
Exhibit B —
From today's Sunday comics:
Exhibit C —
From a Smith student with the
same name as the Princeton student
(i.e., Dagwood's "Twisterooni" twin):
Related illustrations
("Visual Stimuli") from
the Smith student's game —
Literary Exercise:
Continuing the Smith student's
Psychonauts theme,
compare and contrast
two novels dealing with
similar topics:
A Wrinkle in Time,
by the Christian author
Madeleine L'Engle,
and
Psychoshop,
by the secular authors
Alfred Bester and
Roger Zelazny.
Presumably the Princeton student
would prefer the Christian fantasy,
the Smith student the secular.
Those who prefer reality to fantasy —
not as numerous as one might think —
may examine what both 4×4 arrays
illustrated above have in common:
their structure.
Both Princeton and Smith might benefit
from an application of Plato's dictum:
Comments Off on Sunday March 26, 2006
Wednesday, March 15, 2006
Club
From today's Harvard Crimson:
Two members of the Harvard Class of 2007 "scarf down pie at the Math Department's 'pi'-eating contest at 3:14 p.m. yesterday in celebration of Pi Day. Participants had three minutes and 14 seconds to eat as much pie as posssible."
Log24, Feb. 24, 2006:
"What other colleges call fraternities,
Princeton calls Eating Clubs."
Comments Off on Wednesday March 15, 2006
Monday, March 13, 2006
| Le Républicain Lorrain du 14 janvier 2001
Le Lapin Agile veille sur la Butte (par Michel Genson)
24 décembre 1900. Dans son atelier glacial du Bateau Lavoir, à flanc de la colline de Montmartre, Picasso se frotte les yeux. C’est bien Wasley, son ami Wasley, qu’il aperçoit traversant la place Ravignan, courbé sous le poids d’un grand Christ en croix. Le sculpteur titube et s’en va gravissant un à un les escaliers qui mènent au sommet de la Butte, direction la rue des Saules. Car l’œuvre est destinée aux murs du petit estaminet où la bande a trouvé asile, pour y échanger chaque soir des refrains, bocks et vaticinations les plus folles. La bande, c’est à dire Utrillo, Max Jacob, Modigliani et les autres… Un siècle et un souffle de légende plus tard, le même Christ blanc occupe toujours la même place, sous les lumières tamisées du Lapin Agile. À l’abri sous son aisselle droite, l’autoportrait de Picasso en Arlequin a été authentique en son temps. Jusqu’au jour où le grand Frédé, tenancier mythique du lieu, s’est gratté la barbe avant de le céder à un amateur suédois de passage. Depuis l’original a fait le voyage du MAM (Modern Art Museum) de New-York, et la Butte se contente d’une copie.
Pour le reste, rien a changé ou presque pour le doyen des cabarets parisiens. Ni le décor, ni l’esprit. L’incroyable patine noire des murs, posée là par des lustres de tabagies rigolardes ou inspirées, rappelle au générique les voix des habituées de jadis, Apollinaire, Carco, Dullin, Couté, puis Pierre Brasseur, et plus proches de nous encore d’autres débutants, Caussimon, Brassens, François Billetdoux… La liste exhaustive serait impossible à dresser de tous ceux qui ont émargé au livre d’or du Lapin Agile.
« En haut de la rue Saint-Vincent… » La goualante roule sa rime chaotique sur le pavé de la Butte. Au carrefour de la rue des Saules, la façade est avenante, sans apprêts, avec son acacia dans la cour, et cette étrange dénomination, née des amours burlesques entre l’imagination d’un dessinateur et les facéties des usagers. En 1875, André Gill, caricaturiste ami de Rimbaud, croque en effet, pour l’enseigne de l’ancien Cabaret des Assassins, un lapin facétieux sautant d’une marmite. Le temps d’un jeu de mots et le Lapin à Gill gagne son brevet d’agilité. L’épopée commence, que perpétue Yves Mathieu, aujourd’hui propriétaire, mémoire et continuateur d’une histoire somme toute unique. Histoire, qui, pour l’anecdote, faillit se terminer prématurément, sous la pioche des démolisseurs. Vers 1900, les bicoques du maquis montmartrois doivent laisser place à un grand projet immobilier. C’est Aristide Bruant qui sauvera in extremis le cabaret. Il achète l’établissement, laisse Frédé dans les murs qu’il revendra pour « un prix amical » à Paulo, le fils du même Frédé. Lequel Paulo n’est autre que le beau-père de l’actuel patron : « C’est un truc de famille. J’ai commencé à chanter ici en 48, égrène Yves Mathieu. Ensuite j’ai fait de l’opérette à la Gaîté Lyrique, de la revue aux Folies Bergères, je suis parti en Amérique… En revenant j’ai repris le cabaret, ma femme y chante, mes fils sont là, ils apprennent le métier… C’est comme le cirque, c’est le même esprit. »
Malgré les tempêtes et les modes, le Lapin Agile dure et perdure donc. Et sa silhouette pour carte postale inspire toujours les peintres venus de partout. Comme si la halte faisait partie d’un parcours initiatique immuable. Deux pièces pour un minuscule rez-de-chaussée, dans la première, mi-loge, mi-vestiaire, une guitare attend son tour de projecteur. En l’occurrence un faisceau unique clouant le chanteur (l’humoriste ou le diseur) au rideau rouge de la seconde salle. Là où le spectacle se déroule depuis toujours, là où l’on s’accoude sans vergogne à la table d’Apollinaire, sous les lampes toujours drapées de rouge, pour écouter Ferré, Aragon, Mac Orlan ou les rengaines du Folklore populaire montmartrois. Yves Mathieu reste ferme, « ici, pas de sonorisation, pas de haut-parleur. Les gens découvrent la voix humaine. » Un refrain de Piaf glisse jusqu’au « laboratoire », le réduit où les autres artistes du programme dissertent sur l’état du monde. Les meubles de Bruant sont encore là, au hasard d’un coffre breton, un autre de marine, la façade d’un lit clos… « Des trucs d’origine » pour Yves Mathieu, qui malgré les vicissitudes du temps – il s’ingénie toute l’année durant à entretenir un établissement qui ne bénéficie d’aucun classement officiel, ni d’aucun subside – prêche haut et fort sa confiance, « parce qu’on aura plus que jamais besoin de racines, de repères, et qu’ici, c’est tout un pan de patrimoine qu’on défend à travers la chanson française, celle qu’on chante tous ensemble… » Le même secoue sa longue carcasse et se fend d’un sourire entendu : « Quand je descends à Paris, c’est pas pareil. Ici, le jour, c’est comme dans une église. Il y a le silence, et l’impression de ressentir les ondes laissées pare les cerveaux de ces types, là… » Aux murs, dessins ou tableaux laissés par Mac Orlan, Maclet ou Suzanne Valadon jouent avec l’ombre amicale.
Le Lapin Agile, 22 rue des Saules, 75018 Paris. Tel : 01 46 06 85 87
Source:
http://www.au-lapin-agile.com/info4.htm
|
Note the above description
of Christmas Eve 1900,
and the remark that
“Ici, le jour, c’est comme
dans une église.”
A search for more material on
the Wasley Christ leads to
Princeton’s Nassau Church:
|
The fullness of time. I don’t have to call on the physicists among us to conclude that this fullness was not meant to be the end of the time line. That Paul must not have been talking about time in a linear way. Fullness. Complete. Almost perfect. Overflowing with grace. Just right. Fullness. As in “the earth is the Lord’s and the fullness thereof.” Fullness. As in “I pray that you may have the power to comprehend with all of the saints, what is the breadth, and length, and height and depth, and to know the love of Christ that surpasses all knowledge, so that you may be filled with all the fullness of God.” Fullness. As in “For in Christ, all the fullness of God was pleased to dwell.” “When the fullness of time had come, God sent his Son, born of a woman.”
I can remember Christmas Eve as a child….
— Christmas Eve, 2004,
Sermon at Nassau Church by
The Rev. Dr. David A. Davis |
Comments Off on Monday March 13, 2006
Sunday, March 12, 2006
A Circle of Quiet
From the Harvard Math Table page:
“No Math table this week. We will reconvene next week on March 14 for a special Pi Day talk by Paul Bamberg.”
Some friends of mine are in this band.
They’re playing in a bar on Diversey,
way down the bill, around…
I said I’d be there.
Great.
They’re all in the math department.
They’re good.
They have this song called “i.”
You’d like it. Lowercase i.
They just stand there.
They don’t play anything for three minutes.
Imaginary number?
It’s a math joke.
You see why they’re way down the bill. |
From the April 2006 Notices of the American Mathematical Society, a footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:
4 There is a growing literature in the area of postmodern commentaries of [sic] Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems to demonstrate the logical inconsistency of self-government. For a critical view of this and related developments, see Bricmont and Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael Harris’s review of the latter, “I know what you mean!” [9]….
[9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
[13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.
Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:
“Their [Sokal’s and Bricmont’s] philosophy of mathematics, for instance, is summarized in the sentence ‘A mathematical constant like 
doesn’t change, even if the idea one has about it may change.’ ( p. 263). This claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax in Social Text, is criticized by anthropologist Joan Fujimura, in an article translated for IS*. Most of Fujimura’s article consists of an astonishingly bland account of the history of non-euclidean geometry, in which she points out that the ratio of the circumference to the diameter depends on the metric. Sokal and Bricmont know this, and Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s pi”, presumably to avoid trivial objections like Fujimura’s — wasted effort on both sides.32 If one insists on making trivial objections, one might recall that the theorem
that p is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to is injective. In other words, can be identified algebraically with X, the variable par excellence.33
More interestingly, one can ask what kind of object was before the formal definition of real numbers. To assume the real numbers were there all along, waiting to be defined, is to adhere to a form of Platonism.34 Dedekind wouldn’t have agreed.35 In a debate marked by the accusation that postmodern writers deny the reality of the external world, it is a peculiar move, to say the least, to make mathematical Platonism a litmus test for rationality.36 Not that it makes any more sense simply to declare Platonism out of bounds, like Lévy-Leblond, who calls Stephen Weinberg’s gloss on Sokal’s comment ‘une absurdité, tant il est clair que la signification d’un concept quelconque est évidemment affectée par sa mise en oeuvre dans un contexte nouveau!’37 Now I find it hard to defend Platonism with a straight face, and I prefer to regard the formula
as a creation rather than a discovery. But Platonism does correspond to the familiar experience that there is something about mathematics, and not just about other mathematicians, that precisely doesn’t let us get away with saying ‘évidemment’!38
32 There are many circles in Euclid, but no pi, so I can’t think of any other reason for Sokal to have written ‘Euclid’s pi,’ unless this anachronism was an intentional part of the hoax. Sokal’s full quotation was ‘the of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity.’ But there is no need to invoke non-Euclidean geometry to perceive the historicity of the circle, or of pi: see Catherine Goldstein’s ‘L’un est l’autre: pour une histoire du cercle,’ in M. Serres, Elements d’histoire des sciences, Bordas, 1989, pp. 129-149.
33 This is not mere sophistry: the construction of models over number fields actually uses arguments of this kind. A careless construction of the equations defining modular curves may make it appear that pi is included in their field of scalars.
34 Unless you claim, like the present French Minister of Education [at the time of writing, i.e. 1999], that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus would be a physical constant, like the mass of the electron, that can be determined experimentally with increasing accuracy, say by measuring physical circles with ever more sensitive rulers. This sort of position has not been welcomed by most French mathematicians.
35 Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
36 Compare Morris Hirsch’s remarks in BAMS April 94.
37 IS*, p. 38, footnote 26. Weinberg’s remarks are contained in his article “Sokal’s Hoax,” in the New York Review of Books, August 8, 1996.
38 Metaphors from virtual reality may help here.”
* Earlier defined by Harris as “Impostures Scientifiques (IS), a collection of articles compiled or commissioned by Baudouin Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”
What is the moral of all this French noise?
Perhaps that, in spite of the contemptible nonsense at last summer’s Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.
Despite his disdain for Platonism, exemplified in his remarks on the noteworthy connection of pi with the zeta function in the formula given above, Harris has performed a valuable service to mathematics by pointing out the excellent historical work of Catherine Goldstein. Ms. Goldstein has demonstrated that even a French nominalist can be a first-rate scholar. Her essay on circles that Harris cites in a French version is also available in English, and will repay the study of those who, like Barry Mazur and other Harvard savants, are much too careless with the facts of history. They should consult her “Stories of the Circle,” pp. 160-190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).
For the historically-challenged mathematicians of Harvard, this essay would provide a valuable supplement to the upcoming “Pi Day” talk by Bamberg.
For those who insist on limiting their attention to mathematics proper, and ignoring its history, a suitable Pi Day observance might include becoming familiar with various proofs of the formula, pictured above, that connects pi with the zeta function of 2. For a survey, see Robin Chapman, Evaluating Zeta(2) (pdf). Zeta functions in a much wider context will be discussed at next May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).
Comments Off on Sunday March 12, 2006
Friday, March 10, 2006
Women’s History Month continues…
Raiders of the Lost
Comments Off on Friday March 10, 2006
Friday, February 24, 2006
Final Club
For the feast of St. Matthias
(traditional calendar)–
from Amazon.com, a quoted Library Journal review of Geoffrey Wolff‘s novel The Final Club:
“‘What other colleges call fraternities, Princeton calls Eating Clubs. The Final Club is a group of 12 Princeton seniors in 1958 who make their own, distinctive club….
Young adults may find this interesting, but older readers need not join The Final Club.’
— Previewed in Prepub Alert, Library Journal 5/1/90. Paul E. Hutchison, Fisherman’s Paradise, Bellefonte, Pa. Copyright 1990 Reed Business Information, Inc.”
From The Archivist, by Martha Cooley:
“Although I’ve always been called Matt, my first name isn’t Matthew but Matthias: after the disciple who replaced Judas Iscariot. By the time I was four, I knew a great deal about my namesake. More than once my mother read to me, from the New Testament, the story of how Matthias had been chosen by lot to take the place of dreadful Judas. Listening, I felt a large and frightened sympathy for my predecessor. No doubt a dark aura hung over Judas’s chair– something like the pervasive, bitter odor of Pall Malls in my father’s corner of the sofa.
As far as my mother was concerned, the lot of Matthias was the unquestionable outcome of an activity that seemed capricious to me: a stone-toss by the disciples. I tried with difficulty to picture a dozen men dressed in dust-colored robes and sandals, playing a child’s game. One of the Twelve had to carry on, my mother explained, after Judas had perpetrated his evil. The seat couldn’t be left empty. Hence Matthias: the Lord’s servants had pitched their stones, and his had traveled the farthest.”
Comments Off on Friday February 24, 2006
Tuesday, January 24, 2006
ART WARS
for Michael Harris
(See previous entry.)
Related material:A classic book in a postmodern
(“free-floating signs”) cover —
This is my Princeton Companion
to Mathematics, from the days
when Princeton University Press
had higher scholarly standards.
Comments Off on Tuesday January 24, 2006
Monday, January 23, 2006
In Defense of Hilbert
(On His Birthday)
Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:
“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”
Harris adds in a footnote:
“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”
If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”
For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):
[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”
Some of Zach’s references:
[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].
[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990
From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:
“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”
[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….
[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.
[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.
From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:
“What Hilbert wants to secure is not the truth, but the consistency of the old analysis. This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics. In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one. The statements become meaningless figures built up from signs. Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess. Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs. One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess. And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae. By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game. The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules. Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color. Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game. Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge. What is meant to be recognized is that a contradiction can never occur as an end formula of a proof. Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration. One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color. In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game. It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together. Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”
Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:
“Hilbert’s program… received its share of criticism. Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’ But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time? I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous. It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”
— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)
Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:
“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”
One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136. The quotation in the English translation given there:
“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”
Corfield says of this quotation that he’d love to be told the original German. He should consult the above references cited by Richard Zach.
For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.
Comments Off on Monday January 23, 2006
Sunday, January 22, 2006
Comments Off on Sunday January 22, 2006
Friday, December 2, 2005
Proof 101
From a course description:
“This module aims to introduce the student to rigorous university level mathematics….
Syllabus: The idea of and need for mathematical statements and proofs…. proof by contradiction… proof by induction…. the infinite number of primes….”
In the December Notices of the American Mathematical Society, Brian (E. B.) Davies, a professor of mathematics at King’s College London, questions the consistency of Peano Arithmetic (PA), which has the following axioms:
From BookRags.com—
Axiom 1. 0 is a number.
Axiom 2. The successor of any number is a number.
Axiom 3. If a and b are numbers and if their successors are equal, then a and b are equal.
Axiom 4. 0 is not the successor of any number.
Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.
It should be noted that the word “number” as used in the Peano axioms means “non-negative integer.” The fifth axiom deserves special comment. It is the first formal statement of what we now call the “induction axiom” or “the principle of mathematical induction.”
Peano’s fifth axiom particularly troubles Davies, who writes elsewhere:
I contend that our understanding of number should be placed in an historical context, and that the number system is a human invention. Elementary arithmetic enables one to determine the number of primes less than twenty as certainly as anything we know. On the other hand Peano arithmetic is a formal system, and its internal consistency is not provable, except within set-theoretic contexts which essentially already assume it, in which case their consistency is also not provable. The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction, which is an abstraction from our everyday experience….
… Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated. Even today many university students who have been taught the principle of induction prefer to avoid its use, because they do not feel that it is as natural or as certain as a purely algebraic or geometric proof, if they can find one. The feelings of university students may not settle questions about what is truly fundamental, but they do give some insight into our native intuitions.
— E. B. Davies in
“Counting in the real world,”
March 2003 (word format),
To appear in revised form in
Brit. J. Phil. Sci. as
“Some remarks on
the foundations
of quantum mechanics”
Exercise:
Discuss Davies’s claim that
The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction.
Cite the following passage in your discussion.
It will be clear by now that, if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate.
… I can hardly do better than go back to the Greeks. I will state and prove two of the famous theorems of Greek mathematics. They are “simple” theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class. Each is as fresh and significant as when it was discovered– two thousand years have not written a wrinkle on either of them. Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader, however slender his mathematical equipment.
I. The first is Euclid’s proof of the existence of an infinity of prime numbers.
The prime numbers or primes are the numbers
(A) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
which cannot be resolved into smaller factors. Thus 37 and 317 are prime. The primes are the material out of which all numbers are built up by multiplication: thus
666 = 2 . 3 . 3 . 37.
Every number which is not prime itself is divisible by at least one prime (usually, of course, by several). We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.
Let us suppose that it does, and that
2, 3, 5, . . . , P
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number
Q = (2 . 3 . 5 . . . . . P) + 1.
It is plain that Q is not divisible by any of
2, 3, 5, …, P;
for it leaves the remainder 1 when divided by any one of these numbers. But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them. This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.
The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
— G. H. Hardy,
A Mathematician’s Apology,
quoted in the online guide for
Clear and Simple as the Truth:
Writing Classic Prose, by
Francis-Noël Thomas
and Mark Turner,
Princeton University Press
In discussing Davies’s claim that the above proof is by induction, you may want to refer to Davies’s statement that
Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated
and to Hardy’s statement that the above proof is due to Euclid.
Comments Off on Friday December 2, 2005
Wednesday, November 30, 2005
Hobgoblin?
Brian Davies is a professor of mathematics at King’s College London. In the December Notices of the American Mathematical Society, he claims that arithmetic may, for all we know, be inconsistent:
“Gödel taught us that it is not possible to prove that Peano arithmetic is consistent, but everyone has taken it for granted that in fact it is indeed consistent.
Platonistically-inclined mathematicians would deny the possibility that Peano arithmetic could be flawed. From Kronecker onwards many consider that they have a direct insight into the natural numbers, which guarantees their existence. If the natural numbers exist and Peano’s axioms describe properties that they possess then, since the axioms can be instantiated, they must be consistent.”
“It is not possible to prove that Peano arithmetic is consistent”…?!
Where did Gödel say this? Gödel proved, in fact, according to a well-known mathematician at Princeton, that (letting PA stand for Peano Arithmetic),
“If PA is consistent, the formula expressing ‘PA is consistent’ is unprovable in PA.”
— Edward Nelson,
Mathematics and Faith (pdf)
Remarkably, even after he has stated correctly Gödel’s result, Nelson, like Davies, concludes that
“The consistency of PA cannot be concretely demonstrated.”
I prefer the argument that the existence of a model ensures the consistency of a theory.
For instance, the Toronto philosopher William Seager writes that
“Our judgement as to the consistency of some system is not dependent upon that system’s being able to prove its own consistency (i.e. generate a formula that states, e.g. ‘0=1’ is not provable). For if that was the sole basis, how could we trust it? If the system was inconsistent, it could generate this formula as well (see Smullyan,
Gödel’s Incompleteness Theorems, (Oxford, 1992, p. 109)). Furthermore, [George] Boolos allows that we do know that certain systems, such as Peano Arithmetic, are consistent even though they cannot prove their own consistency. Presumably, we know this because we can see that a certain model satisfies the axioms of the system at issue, hence that they are true in that model and so must be consistent.”
— Yesterday’s Algorithm:
Penrose and the Gödel Argument
The relationship between consistency and the existence of a model is brought home by the following weblog entry that neatly summarizes a fallacious argument offered in the AMS Notices by Davies:
The following is an interesting example that I came across in the article “Whither Mathematics?” by Brian Davies in the December issue of Notices of the American Mathematical Society.
Consider the following list A1 of axioms.
(1) There is a natural number 0.
(2) Every natural number a has a successor, denoted by S(a).
(3) There is no natural number whose successor is 0.
(4) Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b).
(5) If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all the natural numbers.
Now consider the following list A2 of axioms.
(1) G is a set of elements and these elements obey the group axioms.
(2) G is finite but not isomorphic to any known list of finite simple groups.
(3) G is simple, in other words, if N is a subset of G satisfying certain properties then N=G.
We can roughly compare A2 with A1. The second axiom in A2 can be thought of as analogous to the third axiom of A1. Also the third axiom of A2 is analogous to the fifth axiom of A1, insofar as it refers to an unspecified set with cetain properties and concludes that it is equal to G.
Now, as is generally believed by most group theorists, the system A2 is internally inconsistent and the proof its inconsistency runs for more than 10000 pages.
So who is to deny that the system A1 is also probably internally inconsistent! Particularly since Godel proved that you can not prove it is consistent (staying inside the system). May be the shortest proof of its inconsistency is one hundred million pages long!
— Posted by Krishna,
11/29/2005 11:46:00 PM,
at his weblog,
“Quasi-Coherent Ruminations”
An important difference between A1 (the set of axioms of Peano arithmetic) and A2 (a set of axioms that describe a new, unknown, finite simple group) is that A1 is known to have a model (the nonnegative integers) and A2 is not known to have a model.
Therefore, according to Seager’s argument, A1 is consistent and A2 may or may not be consistent.
The degree to which Seager’s argument invokes Platonic realism is debatable. Less debatable is the quasireligious faith in nominalism proclaimed by Davies and Nelson. Nelson’s own account of a religious experience in 1976 at Toronto is instructive.
I must relate how I lost my faith in Pythagorean numbers. One morning at the 1976 Summer Meeting of the American Mathematical Society in Toronto, I woke early. As I lay meditating about numbers, I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers, leaving me like an infant in a crib reduced to counting on my fingers. Now I live in a world in which there are no numbers save those that human beings on occasion construct.
— Edward Nelson,
Mathematics and Faith (pdf)
Nelson’s “Mathematics and Faith” was written for the Jubilee for Men and Women from the World of Learning held at the Vatican, 23-24 May 2000. It concludes with an invocation of St. Paul:
During my first stay in Rome I used to play chess with Ernesto Buonaiuti. In his writings and in his life, Buonaiuti with passionate eloquence opposed the reification of human abstractions. I close by quoting one sentence from his
Pellegrino di Roma. “For [St. Paul] abstract truth, absolute laws, do not exist, because all of our thinking is subordinated to the construction of this holy temple of the Spirit, whose manifestations are not abstract ideas, but fruits of goodness, of peace, of charity and forgiveness.”
— Edward Nelson,
Mathematics and Faith (pdf)
Belief in the consistency of arithmetic may or may not be foolish, and therefore an Emersonian hobgoblin of little minds, but bullshit is bullshit, whether in London, in Princeton, in Toronto, or in Rome.
Comments Off on Wednesday November 30, 2005
Wednesday, November 9, 2005
Butterfly Effect
From today’s
online New York Times:
John Fowles on two of his novels:
“I wanted to show the seeds of an intense future evolution in a particular period.”
Ray Bradbury, “A Sound of Thunder” (1952), on the death of a butterfly:
“It fell to the floor, an exquisite thing, a small thing that could upset balances and knock down a line of small dominoes and then big dominoes and then gigantic dominoes, all down the years across Time.”
Zhuangzi:
“Once Zhuang Zhou dreamt he was a butterfly, a butterfly flitting and fluttering around, happy with himself and doing as he pleased. He didn’t know he was Zhuang Zhou. Suddenly he woke up and there he was, solid and unmistakable Zhuang Zhou. But he didn’t know if he was Zhuang Zhou who had dreamt he was a butterfly, or a butterfly dreaming he was Zhuang Zhou. Between Zhuang Zhou and a butterfly there must be some distinction! This is called the Transformation of Things.”
Related material:
Saturday, September 17, 2005
From www.ams.org:
Serge Lang, 1927-2005
“Serge Lang passed away on September 12 at the age of 78. Lang was a professor at Yale University from 1972 to 2005. He received his Ph.D. from Princeton University in 1951 under the direction of Emil Artin. Lang was awarded the Frank Nelson Cole Prize in Algebra in 1960 and the Leroy P. Steele Prize for Mathematical Exposition in 1999. He was well known for his mathematics texts and was a member of the National Academy of Sciences. [Item posted 9/15/05]”
From a review of one of Lang’s books, Challenges:
“Again and again, Lang has caught powerful academics and journalists at evasions, stonewalling, and intimidation. It’s cost him considerable time, effort, and money; it’s also made him a lot of enemies. It should be mentioned here that Professor Lang is also a productive researcher in mathematics and a prolific author of books of mathematics. I literally don’t know how he does it. He must have absolutely no life outside his office.
OK, sure, Lang is a crank. He’s also a national treasure. His commitment to the ethic of honesty and plain speaking should be an example to us all.”
Serge Lang,
May 19, 1927 –
September 12, 2005
Comments Off on Saturday September 17, 2005
Monday, August 22, 2005
The Hole
Part I: Mathematics and Narrative
Apostolos Doxiadis on last month's conference on "mathematics and narrative"–
Doxiadis is describing how talks by two noted mathematicians were related to
"… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "
Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. [In one part] I’ll tell you a few things about proofs. [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’ But the similarity was not clearly explained: the hole, you see, the ‘miracle.’ Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
Part II: Possible Worlds
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer
Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
Background:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)
The God-Shaped Hole
Part III: Modal Theology
"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
— Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)
"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."
"We symbolize
logical necessity
with the box ( )
and logical possibility
with the diamond ( )."
— Keith Allen Korcz
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
|
Comments Off on Monday August 22, 2005
Saturday, June 4, 2005
Drama of the Diagonal The 4×4 Square: French Perspectives
Earendil_Silmarils:
Les Anamorphoses:
“Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d’horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4).”
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
“… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s’opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d’établir
de façon élégante des résultats de
la géométrie élémentaire.”
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
“This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
“… is nothing else but the
relativity theory for the set Sigma,
a set which, by its discrete and
finite character, is conceptually
so much simpler than the
infinite set of points in space
or space-time dealt with
by ordinary relativity theory.”
— Weyl, Symmetry,
Princeton U. Press, 1952
Metaphor and Algebra…
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
— attributed, in varying forms, to
Max Black, Models and Metaphors, 1962
For metaphor and
algebra combined, see
“Symmetry invariance in a diamond ring,”
A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
A New Slant…
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.” The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
Comments Off on Saturday June 4, 2005
Saturday, May 14, 2005
Art at Princeton
From Princeton University Press,
A. W. Mellon Lectures in the
Bollingen Series:
The Bollingen Cross
or “Gnostic wheel”
of Princeton U. Press
Paths to the Absolute:
Mondrian, Malevich, Kandinsky,
Pollock, Newman, Rothko, and Still
by John Golding
Cloth | 2000 | $65.00 | ISBN: 0-691-04896-7
240 pp. | 7 x 10 | 63 color plates 109 halftones
This may illuminate Krauss’s remarks on
Mondrian and Malevich at the
conclusion of the previous entry.
Comments Off on Saturday May 14, 2005
Today’s New York Times:
“Horton Marlais Davies, Putnam professor emeritus of religion at Princeton and an author of many books about church history, died on Wednesday at his home in Princeton, N.J. He was 89…. Dr. Davies specialized in the impact of Christianity on the arts.”
A book edited by Horton Davies,
apparently first published by Eerdmans
at Grand Rapids, Michigan, in 1990
The Catholic Encyclopedia (1908) on the communion of saints:
“One cannot read the parables of the kingdom (Matt., xiii) without perceiving its corporate nature and the continuity which links together the kingdom in our midst and the kingdom to come. The nature of that communion, called by St. John a fellowship with one another (‘a fellowship with us’ — I John, i, 3) because it is a fellowship with the Father, and with his Son, and compared by him to the organic and vital union of the vine and its branches (John, xv), stands out….”
Related material:
Religious art in the entry Art History of 11 AM Wednesday, May 11, the date of Davies’s death. See also the following direct and indirect links from that entry:
To a cruciform artifact from the current film Kingdom of Heaven, to an entry quoting John xv, Nine is a Vine, and to Art Theory for Yom Kippur.
For less-religious material on the number nine, see the entries and links in the Log24 archive for June 17-30, 2004.
From Rosalind Krauss, “Grids”:
“If we open any tract–
Plastic Art and Pure Plastic Art
or The Non-Objective World,
for instance– we will find that
Mondrian and Malevich are not
discussing canvas or pigment or
graphite or any other form of
matter. They are talking about
Being or Mind or Spirit.”
Amen.
Comments Off on Saturday May 14, 2005
Monday, April 4, 2005
Fourth Day of the Fourth Month,
4:04:04
“My wife took, unnoticed, this picture, unposed, of me in the act of writing a novel…. The date (discernible in the captured calendar) is February 27, 1929. The novel, Zashchita Luzhina (The Defense), deals with the defense invented by an insane chess player….”
— Vladimir Nabokov, note to photograph following page 256 in Speak, Memory: An Autobiography Revisited, Vintage International paperback, August 1989
— Quoted in The Matthias Defense
From a site titled Meaning of the Twentieth Century —
“Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called ‘Innovation in Physics,’ he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: ‘We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.’ To that Freeman added: ‘When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!’ “
— Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147
It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson.
— Log24.net, Feast of St. Mark, 2003
See also The Black Queen and The Eight.
In accordance with the theology of the previous entry, based on Zein’s list of the most common Chinese characters, here are some meanings of
character 162:
[si4] {sì} /to watch/to wait/to examine/to spy/
[si4] {sì} /to seem/to appear/similar/like/to resemble/
[si4] {sì} /until/wait for/
[si4] {sì} /rhinoceros indicus/
[si4] {sì} /four/
[si4] {sì} /(surname)/wife of older brother/
[si4] {sì} /Buddhist temple/
[si4] {sì} /6th earthly branch/9-11 a.m./
[si4] {sì} /stream which returns after branching/
[si4] {sì} /place name/snivel/
[si4] {sì} /offer sacrifice to/
[si4] {sì} /hamper/trunk/
[si4] {sì} /plough/ploughshare/
[si4] {sì} /four (fraud-proof)/market/
[si4] {sì} /to feed/
[si4] {sì} /to raise/to rear/to feed/
[si4] {sì} /team of 4 horses/
[si4 bai3 wan4] {sì bǎi wàn} /four million/
[si4 bai3 yi4] {sì bǎi yì} /40 billion/
[si4 cao2] {sì cáo} /feeding trough/
[si4 cao3] {sì cǎo} /forage grass/
[si4 chu4] {sì chù} /all over the place/everywhere and all directions/
[si4 chuan1] {sì chuān} /Sichuan province, China/
[si4 chuan1 sheng3] {sì chuān shěng} /(N) Sichuan, a south west China province/
[si4 de5] {sì de} /seem as if/rather like/
[si4 fang1] {sì fāng} /four-way/four-sided/
[si4 fen1 zhi1 yi1] {sì fēn zhī yī} /one-quarter/
[si4 fu2] {sì fú} /servo/
[si4 fu2 qi4] {sì fú qì} /server (computer)/
[si4 ge4 xiao3 shi2] {sì gè xiǎo shí} /four hours/
[si4 hu5] {sì hu} /apparently/to seem/to appear/as if/seemingly/
[si4 hu5 hen3 an1 quan2] {sì hu hěn ān quán} /to appear (to be) very safe/
[si4 ji1] {sì jī} /to watch for one's chance/
[si4 ji4] {sì jì} /(n) the four seasons/
[si4 liao4] {sì liào} /feed/fodder/
[si4 lun2 ma3 che1] {sì lún mǎ chē} /chariot/
[si4 men2 jiao4 che1] {sì mén jiào chē} /sedan (motor car)/
[si4 mian4 ba1 fang1] {sì miàn bā fāng} /in all directions/all around/far and near/
[si4 mian4 ti3] {sì miàn tǐ} /tetrahedron/
[si4 miao4] {sì miào} /temple/monastery/shrine/
[si4 nian2] {sì nián} /four years/
[si4 nian2 qian2] {sì nián qián} /four years previously/
[si4 nian2 zhi4 de5 da4 xue2] {sì nián zhì de dà xué} /four-year university/
[si4 qian1] {sì qiān} /four thousand/4 000/
[si4 shi2] {sì shí} /forty/40/
[si4 shi2 duo1] {sì shí duō} /more than 40/
[si4 shi2 liu4] {sì shí liù} /forty six/46/
[si4 shi2 san1] {sì shí sān} /43/forty three/
[si4 shi4 er2 fei1] {sì shì ér fēi} /(saying) appeared right but actually was wrong/
[si4 tian1] {sì tiān} /four days/
[si4 xiao4 fei1 xiao4] {sì xiào fēi xiào} /(saying) resemble a smile yet not smile/
[si4 xue3] {sì xuě} /snowy/
[si4 yang3] {sì yǎng} /to raise/to rear/
[si4 yang3 zhe3] {sì yǎng zhě} /feeder/
[si4 yuan4] {sì yuàn} /cloister/
[si4 yue4] {sì yuè} /April/fourth month/
[si4 yue4 shi2 qi1 hao4] {sì yuè shí qī hào} /April 17/
[si4 zhi1] {sì zhī} /(n) the four limbs of the body/
[si4 zhou1] {sì zhōu} /all around/
— ktmatu.com Chinese-English dictionary
Comments Off on Monday April 4, 2005
Friday, March 11, 2005
| Philadelphia Story
Friday, March 11, 2005
James Biddle, 75, died at his family’s estate, Andalusia, near Philadelphia yesterday.
From today’s New York Times:
“Mr. Biddle was born at Andalusia on July 8, 1929. He attended St. Paul’s School in Concord, N.H., and graduated from Princeton, where he studied art and archaeology. He served in an intelligence unit of the Army during the Korean War.”
“Anybody invites you to a game of solitaire, you tell ’em sorry, buster, the ball game is over.”
— Frank Sinatra |
|
Comments Off on Friday March 11, 2005
Sunday, February 20, 2005
Relativity Blues
Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry. Here is some related material.
In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).
(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)
On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:
"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."
One possible response to such statements was recently provided in some detail by a Princeton philosophy professor. See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.
A more thoughtful response would take into account the following:
1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson
2. The physics underlying Einstein's remarks on free will, God, and dice
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels Secret Passages and Broken Symmetries
4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University
5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.
From a review of Gilead by Jane Vandenburgh:
"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"
For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries. If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003, and those of November 5, 2002.
From Many Dimensions (Eerdmans paperback, 1963, page 53):
"Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?"
A recent answer:
Modal Theology
"We symbolize logical necessity
with the box ()
and logical possibility
with the diamond ()."
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by 
?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Comments Off on Sunday February 20, 2005
Tuesday, February 15, 2005
Answer
“Are you now, or have you ever been?”
“In the case of the Cartesian question, the answer is affirmative, and metaphysics has produced, in the four hundred years since, nothing much better than this. It is not only interesting but supremely practical. What could be more useful than having the means of convincing oneself that one exists whenever the question should arise?”
— Rebecca Goldstein,
Properties of Light
“… a nightshirted boy trying desperately to awake from the iridescent dizziness of dream life. Its ultimate vision was the incandescence of a book or a box grown completely transparent and hollow. This is, I believe, it : not the crude anguish of physical death but the incomparable pangs of the mysterious mental maneuver needed to pass from one state of being to another.”
“Le terme que l’on traduit par dédicace est en japonais ekô, littéralement ‘se tourner vers’. Il est composé de deux idéogrammes, e qui signifie ‘tourner le dos, se tourner, revenir en arrière’ et kô, ‘faire face, s’adresser à’.”
Comments Off on Tuesday February 15, 2005
Tuesday, February 8, 2005
which is (1+Sqrt(5))/2,
about 1.618034.
The Greek letter ‘phi’
(see below for the symbol)
is sometimes used
to represent this number.”
— Don Cohen
For another approach to
the divine proportion, see
Best Picture.
“The
rogue’s yarn that will run through much of the material is the algebraic symmetry to which the name of Galois is attached and which I wanted to introduce in as concrete and appealing a way as possible….
Apart from its intrinsic appeal, that is the reason for treating the construction of the pentagon, and our task today will be to acquire some feel for this construction. It is not easy.”
— R. P. Langlands, 1999 lecture (pdf) at the Institute for Advanced Study, Princeton, in the spirit of Hermann Weyl
Comments Off on Tuesday February 8, 2005
Monday, January 24, 2005
Old School Tie
“We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates’ yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, ‘There must be a formula for how ugly your tie is.’ ”
“Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections……. diamonds and boxes are upper and lower adjoints of Galois connections….”
Evariste Galois
and Log24.net, May 20, 2004:
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962
For metaphor and
algebra combined, see
“Symmetry invariance
in a diamond ring,”
A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
Tuesday, September 28, 2004
3:33:33 PM
Romantic Interaction, continued…
The Rhyme of Time
From American Dante Bibliography for 1983:
|
Freccero, John. "Paradiso X: The Dance of the Stars" (1968). Reprinted in Dante in America … (q.v.), pp. 345-371. [1983]
Freccero, John. "The Significance of terza rima." In Dante, Petrarch, Boccaccio: Studies in the Italian Trecento … (q.v.), pp. 3-17. [1983]
Interprets the meaning of terza rima in terms of a temporal pattern of past, present, and future, with which the formal structure and the thematics of the whole poem coordinate homologically: "both the verse pattern and the theme proceed by a forward motion which is at the same time recapitulary." Following the same pattern in the three conceptual orders of the formal, thematical, and logical, the autobiographical narrative too is seen "as forward motion that moves towards its own beginning, or as a form of advance and recovery, leading toward a final recapitulation." And the same pattern is found especially to obtain theologically and biblically (i.e., historically). By way of recapitulation, the author concludes with a passage from Augustine's Confessions on the nature of time, which "conforms exactly to the movement of terza rima." Comes with six diagrams illustrating the various patterns elaborated in the text.
|
From Rachel Jacoff's review of Pinsky's translation of Dante's Inferno:
"John Freccero's Introduction to the translation distills a compelling reading of the Inferno into a few powerful and immediately intelligible pages that make it clear why Freccero is not only a great Dante scholar, but a legendary teacher of the poem as well."
From The Undivine Comedy, Ch. 2, by Teodolinda Barolini (Princeton University Press, 1992):
|
"… we exist in time which, according to Aristotle, "is a kind of middle-point, uniting in itself both a beginning and an end, a beginning of future time and an end of past time."* It is further to say that we exist in history, a middleness that, according to Kermode, men try to mitigate by making "fictive concords with origins and ends, such as give meaning to lives and to poems." Time and history are the media Dante invokes to begin a text whose narrative journey will strive to imitate– not escape– the journey it undertakes to represent, "il cammin di nostra vita."
* Aristotle is actually referring to the moment, which he considers indistinguishable from time: "Now since time cannot exist and is unthinkable apart from the moment, and the moment is a kind of middle-point, uniting as it does in itself both a beginning and an end, a beginning of future time and an end of past time, it follows that there must always be time: for the extremity of the last period of time that we take must be found in some moment, since time contains no point of contact for us except in the moment. Therefore, since the moment is both a beginning and an end there must always be time on both sides of it" (Physics 8.1.251b18-26; in the translation of R. P. Hardie and R. K. Gaye, in The Basic Works of Aristotle, ed. Richard McKeon [New York: Random House, 1941]).
|
From Four Quartets:
And the pool was filled with water out of sunlight,
And the lotos rose, quietly, quietly,
The surface glittered out of heart of light,
And they were behind us, reflected in the pool.
Then a cloud passed, and the pool was empty.
Go, said the bird, for the leaves were full of children,
Hidden excitedly, containing laughter.
Go, go, go, said the bird: human kind
Cannot bear very much reality.
Time past and time future
What might have been and what has been
Point to one end, which is always present.
Comments Off on Tuesday September 28, 2004
Wednesday, July 28, 2004
The Freshmen, Part II
From the Daily Princetonian,
Feb. 3, 2004:
Caption: Cate Edwards' Princeton friends support her and her father.
"… when Sen. John Edwards, father of Cate Edwards '04, decided to run for president, the troop of 17 students sacrificed tans and theses to pile into a fleet of minivans headed to New Hampshire….
These volunteers… were on a first name basis with the man who had helped them move into freshman dorm rooms and had discussed Senate votes with them over Chinese food."
Log24 May 22, 2004:
From Chuck Polisher's
I Ching Lexicon:
"It's claimed that
if you take a mirror
and look backwards
into a well,
you'll see your future
down in the water."
— Cold Mountain,
Vintage paperback, 1998,
page 48
"Goin' to Carolina in my mind…"
— James Taylor
Comments Off on Wednesday July 28, 2004
Monday, July 12, 2004
Character and Values
In response to this morning’s Wizard-of-Id example (see 1:22 PM entry) of a political Bob-Hope-style Christian wisecrack (a style more apt to make me gag than laugh), some further quotations:
I need a photo-opportunity,
I want a shot at redemption.
Don’t want to end up a cartoon
In a cartoon graveyard.
— Paul Simon
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|
The Washington Post on the gigolo candidate in Boston Monday:
“In a lunch speech to more than 1,000 women who had donated $500 to $2,000 to his campaign or the Democratic Party, Kerry was joined on stage by his wife, Teresa Heinz Kerry…. He focused his comments on improving health care and creating more jobs — notions that he said ‘are not Democratic values. They’re not Republican values. They are American values.’ “
Let us pass over Kerry’s ignorance of the difference between desiderata (things considered desirable) and values (principles, standards, or qualities considered desirable).
A definition of “values” in a different sense, one that might appeal to the late St. Laurance Rockefeller, dead on 7/11, who majored in philosophy at Princeton:
“In an artistical composition, the character of any one part in its relation to other parts and to the whole — often used in the plural: as, the values are well given, or well maintained.”
— Webster’s Unabridged Dictionary, 1913
Rockefeller is, I hope, now in a place where he can discuss this definition with Bach as it applies to, say, that composer’s “Goldberg Variations.”
Here below, another sort of Goldberg Variations seems appropriate to the times we live in …
The following composition was inspired by Whoopi Goldberg’s remarks at last Thursday’s Radio City Music Hall Democratic Party fund-raiser.
Democratic Political Art:
Motherhood and Apple Pie
Sources:
Ike Turner, Bad Dreams album,
Mom’s Apple Pie album (X-rated),
and Log24 entries of
July 9-10 and July 12.
Update of 3:17 AM July 13, 2004:
A place in Heaven next to St. Laurance
seems to have been reserved:
Comments Off on Monday July 12, 2004
Monday, May 31, 2004
Language Games:
Now You’re Playing
with Power
My latest preoccupation…
Using search-and-replace programs to reformat earlier Xanga entries. This involves the use of “regular expressions,” which lead to the following thoughts….
It seems that pure mathematics (i.e., the theory of finite automata) is not without relevance even in very practical data formatting problems. One of the first math books I ever
bought — perhaps the very
first —was
Automata Studies (Princeton’s
Annals of Mathematics Studies, No. 34, 1956). This book, which I still have, begins with an essay by
Stephen Cole Kleene.
Kleene’s legacy includes regular expressions and Kleene’s theorem. For further details, see
Notes on
Formal Language Theory
and Parsing
James Power
Department of Computer Science
NATIONAL UNIVERSITY OF IRELAND, MAYNOOTH
Maynooth, Co. Kildare, Ireland.
Click on picture
for details.
Here’s more on
language games and robot wisdom
from an authority on James Joyce,
Friday, May 14, 2004
Moral Hazard —
The Devil and Wallace Stevens,
continued from May 1-2 entries:
Law Day,
Readings for Law Day,
Fallen from Heaven, and
The Script
University of Southern California, Department of Economics — Industrial Organization ECN 680 – Autumn 2002 — Introduction to Contract and Organization Theory —
Professor Jean-Jacques Laffont
(September 4 – October 21):
“The purpose of this course is to introduce the student to modern contract and organization theory. Part 1 of the course focuses upon models with moral hazard and adverse selection.”
From the insurance page at
http://ingrimayne.saintjoe.edu/:
“The size of the insurance industry indicates that people are eager to pay to avoid risk. They pay and get nothing if fortune smiles on them, whereas if misfortune strikes, they break even because the insurance should just pay back the value lost in the misfortune.
Sometimes, however, people do better than break even when misfortune strikes, and this possibility has greatly interested economists. If, for example, the misfortune costs a person $1000, but insurance will pay $2000, the insured person has no incentive to avoid the misfortune and may act to bring it on. This tendency of insurance to change behavior is called moral hazard.
Sometimes moral hazard is dramatic….
People who know that they face large risks are more likely to buy insurance than people who face small risks. Insurance companies try to minimize the problem that only the people with big risks will buy their product, which is the problem of adverse selection ….”
From today’s New York Times:
“Jean-Jacques Laffont, an economist known for developing mathematical models to estimate what something is worth in situations of deep uncertainty, died on May 1 in Toulouse, France. He was 57….
…Jerry R. Green of Harvard said he was ‘an architect of systems’ and ‘a very original figure.’
Eric Maskin, a professor at the Institute for Advanced Study in Princeton, N.J., called Dr. Laffont ‘simply one of the major figures of our time.’
‘Many people would say he was the leading economist in Europe,’ he added, ‘and that wouldn’t be an unfair judgment.’
Although Dr. Laffont’s models were abstruse enough to satisfy the most theoretical economists, Dr. Green said they were adapted for practical purposes by companies, as well as by public television for scheduling programs.”
Tuesday, April 20, 2004
Rhetorical Question
Yesterday's Cartesian theatre continues….
Robert Osserman, a professor emeritus of mathematics at Stanford University, is special-projects director at the Mathematical Sciences Research Institute, in Berkeley, Calif.
Osserman at aldaily.com today:
"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales — regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all — into fascinating royalty, portrayed on stage and screen….
Who bestowed the magic kiss on the mathematical frog?"
Answer:
William Randolph Hearst III.
"Trained as a mathematician at Harvard, he now likes to hang out with Ken Ribet and the other gurus at the University of California, Berkeley's prestigious Mathematical Sciences Research Institute. Two years ago, he moderated a panel of math professors discussing Princeton professor Andrew Wiles's historic proof of Fermat's Last Theorem."
— Wired magazine, June 1995
See also
Hearst Gift Spurs Math Center Expansion and
Review of Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate (pdf), Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 248–252,
by William Randolph Hearst III
and Kenneth A. Ribet.
Chet Atkins summarizes:
"And that's the secret of frog kissin', and you can do it too if you'll just listen.
Just slow down, turn around, bend down and kiss you a frog! Ribet! Ribet!"
Comments Off on Tuesday April 20, 2004
Friday, March 5, 2004
Songs for Shakespeare
from Willie and Waylon
|
From today's New York Times
by Ben Brantley
…."Dost thou know me, fellow?" thunders Christopher Plummer, who is giving the performance of a lifetime in the title role of "King Lear"….
Throughout Jonathan Miller's engrossing production of Shakespeare's bleakest tragedy, which opened last night, Mr. Plummer bestrides the boundary between being and nothingness….
The Line,
by S.H. Cullinane
|
LEAR:
Now you better do some thinkin'
then you'll find
You got the only daddy
that'll walk the line.
FOOL:
I've always been different
with one foot over the line….
I've always been crazy
but it's kept me from going insane.
|
FOOL:
174. …. Now thou art an 0 without
175. a figure. I am better than thou art, now. I am a fool;
176. thou art nothing….
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"…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead."
— The Greater Trumps,
by Charles Williams, Ch. 14
Follow-up of Friday, March 5
From Arts & Letters Daily,
Weekend Edition, March 6-7, 2004 —
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Some readers crave awe more than understanding, and lurid pop science is always there to feed their addiction to junk ideas… more»
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Does Shakespeare’s Lear have a spiritual dimension? “No,” insists Jonathan Miller. “That’s modern, New Age drivel…." more»
|
The "more" link of the item at left above leads to an American Scientist article titled
The Importance of
Being Nothingness.
The appearance of these two items side-by-side at Arts & Letters Daily, together with Brantley's remark above, is an example of Jungian synchronicity — a concept that the American Scientist author and Jonathan Miller probably both sneer at. Sneer away.
Comments Off on Friday March 5, 2004
Wednesday, March 3, 2004
Deep Play
In the previous entry, there was a reference to Carl Kaysen, former director of the Institute for Advanced Study at Princeton and father of Susanna Kaysen, author of Girl, Interrupted.
A search for further information on Carl Kaysen led to
Mark Turner, Cognitive Dimensions of Social Science: The Way We Think About Politics, Economics, Law, and Society, Oxford University Press, 2001. For a draft of this work, click here.
Turner's book describes thought and culture in terms of what he calls "blends." It includes a meditation on
Clifford Geertz, "Deep Play: Notes on the Balinese Cockfight," in Dædalus, Journal of the American Academy of Arts and Sciences, issue entitled, "Myth, Symbol, and Culture," Winter 1972, volume 101, number 1
That Turner bases weighty ruminations of what he is pleased to call "social science" on the properties of cockfights suggests that the academic world is, in some respects, even more bizarre than the mental hospital described by Kaysen's daughter.
Still, Turner's concept of "blends" is not without interest.
Here is a blend based on a diagram of the fields in which Turner and Kaysen père labor:
"politics, economics,
law, and society" (Turner)
and "economics, sociology,
politics and law" (Kaysen).
In the previous entry we abstracted from the nature of these academic pursuits, representing them simply as sets in a Venn diagram. This led to the following religious icon, an example of a Turner "blend" —
The Jewel
in Venn's Lotus.
Here is another "blend," related both to the religious material in the previous entry and to Geertz's influential essay.
From my entry for
St. Patrick's Day, 2003:
Summa Theologica
How can you tell there's an Irishman
present at a cockfight?
He enters a duck.
How can you tell a Pole is present?
He bets on the duck.
How can you tell an Italian is present?
The duck wins.
(Source: Blanche Knott,
Truly Tasteless Jokes)
Illustration for the entries
of Oct. 27, 2003:
El Pato-lógico and a
"dream of heaven."
Comments Off on Wednesday March 3, 2004
Friday, February 20, 2004
Finite Relativity
Today is the 18th birthday of my note
“The Relativity Problem in Finite Geometry.”
That note begins with a quotation from Weyl:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.
Tuesday, February 3, 2004
Retiring Faculty
The following is related to
today's previous four log24 entries.
From my paper journal, a Xeroxed note, composed entirely of cut copies
of various documents,
from July 11, 1990….
|
Harvard Alumni Gazette June 1990
Retiring Faculty Continue their Love of Learning, Creativity
Thought for today: "He who tells the truth must have one foot in the stirrup." — Armenian Proverb
Preserve me from the enemy
who has something to gain: and
from the friend who has something to lose.
Remembering the words of
Nehemiah the Prophet:
"The trowel in hand, and the gun
rather loose in the holster."
— T. S. Eliot, Choruses from the Rock — 1934
Pattern in Islamic Art is the most thorough study yet published of the structure of the art.
|
Oleg Grabar, Aga Khan Professor of Islamic Art, will join the Institute for Advanced Study at Princeton, where he will devote himself to pure research. He has three books planned — which he estimates will take him about four years to finish — including books on the theory of ornament, and studies of early medieval Jerusalem and Islamic Sicily. "I'm also planning to brush up on my Persian, which I had kind of forgotten," he said.
|
Clint Eastwood is the nameless stranger who mysteriously appears in the Warner Brothers film 'Pale Rider.'
|
Closing the cylinder, he holstered the gun, pivoted, and strode across the now silent street toward his horse.
An ashen-faced Lahood stared out the second-story window, following the tall man's movements. In his right hand he held a long-barreled blue-black derringer. He raised the muzzle purposefully.
The Preacher put a foot in the stirrup and hesitated. Turning, he lifted his eyes to a particular window. The curtains behind it moved slightly. The report of the single shot was muffled by distance and glass. From his position the Preacher could not hear the thump of the body as it struck the thick Persian rug. He did not have to hear it.
Lahood had begun this day's work, and Lahood had finished it.
|
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Sources: Harvard Alumni Gazette, local newspaper, a volume of the poems of T. S. Eliot, David Wade's Pattern in Islamic Art, and a paperback novelization of Pale Rider
Saturday, December 20, 2003
White, Geometric, and Eternal
This afternoon's surfing:
Prompted by Edward Rothstein's own Fides et Ratio encyclical in today's NY Times, I googled him.
At the New York Review of Books, I came across the following by Rothstein:
"… statements about TNT can be represented within TNT: the formal system can, in a precise way, 'talk' about itself."
This naturally prompted me to check what is on TNT on this, the feast day of St. Emil Artin. At 5 PM this afternoon, we have Al Pacino in "The Devil's Advocate" — a perfect choice for the festival of an alleged saint.
Preparing for Al, I meditated on the mystical significance of the number 373, as explained in Zen and Language Games: the page number 373 in Robert Stone's theological classic A Flag for Sunrise conveys the metaphysical significance of the phrase "diamonds are forever" — "the eternal in the temporal," according to Stone's Catholic priest. This suggests a check of another theological classic, Pynchon's Gravity's Rainbow. Page 373 there begins with the following description of prewar Berlin:
"white and geometric."
This suggests the following illustration of a white and geometric object related to yesterday's entry on Helmut Wielandt:
From antiquark.com
Figure 1
(This object, which illustrates the phrase "makin' the changes," also occurs in this morning's entry on the death of a jazz musician.)
A further search for books containing "white" and "geometric" at Amazon.com yields the following:
Figure 2
From Mosaics, by
Fassett, Bahouth, and Patterson:
"A risco fountain in Mexico city, begun circa 1740 and made up of Mexican pottery and Chinese porcelain, including Ming.
The delicate oriental patterns on so many different-sized plates and saucers [are] underlined by the bold blue and white geometric tiles at the base."
Note that the tiles are those of Diamond Theory; the geometric object in figure 1 above illustrates a group that plays a central role in that theory.
Finally, the word "risco" (from Casa del Risco) associated with figure 2 above leads us to a rather significant theological site associated with the holy city of Santiago de Compostela:
Figure 3
Vicente Risco's
Dedalus in Compostela.
Figure 3 shows James Joyce (alias Dedalus), whose daughter Lucia inspired the recent entry Jazz on St. Lucia's Day — which in turn is related, by last night's 2:45 entry and by Figure 1, to the mathematics of group theory so well expounded by the putative saint Emil Artin.
"His lectures are best described as
polished diamonds."
— Fine Hall in its Golden Age,
by Gian-Carlo Rota
If Pynchon plays the role of devil's advocate suggested by his creation, in Gravity's Rainbow, of the character Emil Bummer, we may hope that Rota, no longer in time but now in eternity, can be persuaded to play the important role of saint's advocate for his Emil.
Update of 6:30 PM 12/20/03:
Riddled:
The Absolutist Faith
of The New York Times
White and Geometric, but not Eternal.
Comments Off on Saturday December 20, 2003
Thursday, November 6, 2003
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.
Comments Off on Thursday November 6, 2003
Friday, October 10, 2003
Pro-Semitism
As the president of Harvard has pointed out, many have found good reasons recently to become anti-Semites.
Here are three good reasons to be pro-Semitic:
- Jordan Ellenberg’s course on Numbers, Equations, and Proofs this fall at Princeton,
- Ellenberg’s phrase, contained in a syllabus for that course (3-7 Nov. 2003):
“The heartbreaking failure of unique factorization,” and
- Ellenberg himself.
Unlike many with a Harvard background who project with success the appearance of intelligence, Ellenberg seems to be in fact genuinely intelligent… a rare thing.
He also seems to be Jewish… This may be false, though, since Elllenberg, intelligently, does not state any ethnic or religious preferences.
Fine Hall 1201
The classic question of Sir Laurence Olivier– “Is it safe?”– may, in view of the above, be answered in the affirmative… provided, that is, that the “it” refers to number theory at Princeton… one of the crowning glories of Western civilization.
Comments Off on Friday October 10, 2003
Saturday, August 23, 2003
Pictures of Nothing
‘”The artist delights to go back to the first chaos of the world… All is without forms and void. Some one said of his landscapes that they were pictures of nothing, and very like.”
— William Hazlitt, 1816, on J. M. W. Turner
“William Hazlett [sic] once described Turner’s painting as ‘pictures of the elements of air, earth, and water. The artist delights to go back to the first chaos of the world…All is without form and void. Some one said of his landscapes that they were pictures of nothing and very like.‘ This description could equally well be applied to a Pollock, Newman, or Rothko.”
— Sonja J. Klein, thesis, The Nature of the Sublime, September 2000
The fifty-second A. W. Mellon series of Lectures in the Fine Arts was given last spring at the National Gallery in Washington, D.C., by Kirk Varnedoe, art historian at the Institute for Advanced Study, Princeton, New Jersey.
| The lecture series was titled
Pictures of Nothing:
Abstract Art since Pollock.
The lectures, 2003:
Why Abstract Art? … March 30
Survivals and Fresh Starts … April 6
Minimalism … April 13
After Minimalism … April 27
Satire, Irony, and Abstract Art … May 4
Abstract Art Now … May 11 |
Varnedoe died on Thursday, August 14, 2003,
the day of the Great Blackout.
Pictures of Nothing:
“Record-breaking crowds turned up at the National Gallery for Kirk’s Mellon Lectures….
… the content of Kirk’s talk was miraculously subtle, as he insisted that there could be no single explanation for how abstraction works, that each piece had to be understood on its own terms — how it came to be made, what it meant then and what it has gone on to mean to viewers since.
Dour works like
Frank Stella’s early
gray-on-black canvases …
“Die Fahne Hoch,”
Frank Stella,
1959
“Gray on Black,”
or “Date of Death”
seemed to open up under Kirk’s touch to reveal a delicacy and complexity lost in less textured explanations.”
— Blake Gopnik in the Washington Post,
Aug. 15, 2003
For another memorial to Varnedoe, see
Fahne Hoch.
A May 18 Washington Post article skillfully summarized Varnedoe’s Mellon Lectures at the National Gallery:
Closing the Circle on Abstract Art.
For more on art and nihilism, see
The Word in the Desert.
Comments Off on Saturday August 23, 2003
Sunday, July 13, 2003
ART WARS, 5:09
The Word in the Desert
For Harrison Ford in the desert.
(See previous entry.)
Words strain,
Crack and sometimes break,
under the burden,
Under the tension, slip, slide, perish,
Will not stay still. Shrieking voices
Scolding, mocking, or merely chattering,
Always assail them.
The Word in the desert
Is most attacked by voices of temptation,
The crying shadow in the funeral dance,
The loud lament of
the disconsolate chimera.
— T. S. Eliot, Four Quartets
The link to the word "devilish" in the last entry leads to one of my previous journal entries, "A Mass for Lucero," that deals with the devilishness of postmodern philosophy. To hammer this point home, here is an attack on college English departments that begins as follows:
"William Faulkner's Snopes trilogy, which recounts the generation-long rise of the drily loathsome Flem Snopes from clerk in a country store to bank president in Jefferson, Mississippi, teems with analogies to what has happened to English departments over the past thirty years."
For more, see
The Word in the Desert,
by Glenn C. Arbery.
See also the link on the word "contemptible," applied to Jacques Derrida, in my Logos and Logic page.
This leads to an National Review essay on Derrida,
The Philosopher as King,
by Mark Goldblatt.
A reader's comment on my previous entry suggests the film "Scotland, PA" as viewing related to the Derrida/Macbeth link there.
I prefer the following notice of a 7-11 death, that of a powerful art museum curator who would have been well cast as Lady Macbeth:
From the Whitney Museum site:
"Max Anderson: When artist Frank Stella first showed this painting at The Museum of Modern Art in 1959, people were baffled by its austerity. Stella responded, 'What you see is what you see. Painting to me is a brush in a bucket and you put it on a surface. There is no other reality for me than that.' He wanted to create work that was methodical, intellectual, and passionless. To some, it seemed to be nothing more than a repudiation of everything that had come before—a rational system devoid of pleasure and personality. But other viewers saw that the black paintings generated an aura of mystery and solemnity.
The title of this work, Die Fahne Hoch, literally means 'The banner raised.' It comes from the marching anthem of the Nazi youth organization. Stella pointed out that the proportions of this canvas are much the same as the large flags displayed by the Nazis.
But the content of the work makes no reference to anything outside of the painting itself. The pattern was deduced from the shape of the canvas—the width of the black bands is determined by the width of the stretcher bars. The white lines that separate the broad bands of black are created by the narrow areas of unpainted canvas. Stella's black paintings greatly influenced the development of Minimalism in the 1960s."
From Play It As It Lays:
She took his hand and held it. "Why are you here."
"Because you and I, we know something. Because we've been out there where nothing is. Because I wanted—you know why."
"Lie down here," she said after a while. "Just go to sleep."
When he lay down beside her the Seconal capsules rolled on the sheet. In the bar across the road somebody punched King of the Road on the jukebox again, and there was an argument outside, and the sound of a bottle breaking. Maria held onto BZ's hand.
"Listen to that," he said. "Try to think about having enough left to break a bottle over it."
"It would be very pretty," Maria said. "Go to sleep."
I smoke old stogies I have found…
Cigar Aficionado on artist Frank Stella:
" 'Frank actually makes the moment. He captures it and helps to define it.'
This was certainly true of Stella's 1958 New York debut. Fresh out of Princeton, he came to New York and rented a former jeweler's shop on Eldridge Street on the Lower East Side. He began using ordinary house paint to paint symmetrical black stripes on canvas. Called the Black Paintings, they are credited with paving the way for the minimal art movement of the 1960s. By the fall of 1959, Dorothy Miller of The Museum of Modern Art had chosen four of the austere pictures for inclusion in a show called Sixteen Americans."
For an even more austere picture, see
Geometry for Jews:
For more on art, Derrida, and devilishness, see Deborah Solomon's essay in the New York Times Magazine of Sunday, June 27, 1999:
How to Succeed in Art.
"Blame Derrida and
his fellow French theorists…."
See, too, my site
Art Wars: Geometry as Conceptual Art.
For those who prefer a more traditional meditation, I recommend
Ecce Lignum Crucis
("Behold the Wood of the Cross")
THE WORD IN THE DESERT
For more on the word "road" in the desert, see my "Dead Poet" entry of Epiphany 2003 (Tao means road) as well as the following scholarly bibliography of road-related cultural artifacts (a surprising number of which involve Harrison Ford):
A Bibliography of Road Materials
Tuesday, July 1, 2003
Jew’s on First
This entry is dedicated to those worshippers of Allah who have at one time or another cried
“Itbah al-Yahud!” … Kill the Jew!
(See June 29 entries).
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Dead at 78
Comedian Buddy Hackett died on Tuesday, July First, 2003, according to the New York Times. According to Bloomberg.com, he died Sunday or Monday. |
Associated Press
Buddy Hackett,
on the set of
“It’s a Mad, Mad,
Mad, Mad World”
in 1962. |
| Whatever. We may imagine he has now walked, leading a parade of many other stand-up saints, into a bar. |
Hepburn at Chaillot
|
|
MIDRASH
for Buddy Hackett
From my May 25 entry,
Matrix of the Death God:
R. M. Abraham’s Diversions and Pastimes, published by Constable and Company, London, in 1933, has the following magic square:
The Matrix of Abraham
A summary of the religious import of the above from Princeton University Press:
“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”
— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72
Other appearances of this religious icon on the Web include:
On Linguistic Creation
Picasso’s Birthday
1991 Yearbook
Rolling Stone
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Hackett
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In the Picasso’s Birthday version, 22 of the 25 magic square cells are correlated with pictures on the “Class of ’91” cover of Rolling Stone magazine. Number 7 is Rod† Stewart. In accordance with the theological rhyme “Seven is heaven, eight is a gate,” our site music for today is “Forever Young,” a tune made famous by Stewart.
† Roderick, actually — the name of the hero in “Madwoman of Chaillot”
Comments Off on Tuesday July 1, 2003
Friday, June 13, 2003
Born on this date:
William Butler Yeats.
“Surely some revelation
is at hand” — W. B. Yeats
Behold a Pale Horse:
A link in memory of Gregory Peck.
In Slouching Towards Bethlehem,
Joan Didion wrote that
“The oral history of Los Angeles
is written in piano bars.”
Today’s site music, a piano rendition of “Speak Low,” from “One Touch of Venus,” was suggested by
- the “black triangle” theme of Wednesday’s entry and by
- the name “Amy Hollywood.”
Ms. Hollywood has an essay in the April 2003 Princeton journal Theology Today.
My own theological interests (besides those expressed in the “black triangle” link above) are much closer to those in a 2001 First Things essay, The End of Endings.
Washington Square Press paperback, 1981, page 222
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Friday, June 6, 2003
SKEPTIC, or
Beware of…
Jews Telling Stories
“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday.”
— Bernard Holland, C12, N.Y. Times, 5/20/96
From my entry of Monday, June 2, 2003:
William Holden and Martha Scott
in “Our Town,” 1940
Holden Scott
From a website titled Child of a Voice:
“The Talmud says, even in a time when there is no more prophecy there can still be the Daughter of a Voice.
The Tosefot explain: this is like the sound the echo of a hammer makes when it strikes something, and the sound echoes back from mountains. Not the Voice, but a daughter, a child of it.
Not the sound but the echo of a sound. Not the prophecy from God in its purest way, but in a less pure way.
Now because of our sins there is no more prophecy but in a time when there is no prophecy there can be Daughter of a Voice.”
Copyright Abraham Mezrich 2003
From a July 1999 review of a novel:
“The good news is that this is perhaps Ben Mezrich’s finest thriller. The irony is that he used a pen name on it.”
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The
novel is
Skeptic;
the author,
Ben Mezrich,
used the pen name
Holden Scott.
From an interview
with Ben Mezrich:
“Mezrich, the author of several critically acclaimed thrillers, came to Boston from Princeton, New Jersey, by way of Harvard University, where he graduated – magna cum laude, mind you – in 1991…. In his Boston apartment…. prominently exhibited was a paperback biography of local boy made good Matt Damon.”
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Comments Off on Friday June 6, 2003
Sunday, May 25, 2003
— ART WARS —
Mental Health Month, Day 25:
Matrix of the Death God
Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham. (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.) For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.
See, too, The Matrix of Abraham, illustrated below. This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.
The Matrix of Abraham
A summary of the religious import of the above from Princeton University Press:
“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”
— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72
Other appearances of this religious icon on the Web:
On Linguistic Creation
Picasso’s Birthday
A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).
This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.
Thursday, May 22, 2003
Mental Health Month:
Springtime for Wagner
“And now what you’ve all been waiting for…
Wagner!“
— Colin Hay as Zac in the film “Cosi“
“When I sought those who would sympathize with my plans, I had only you, the friends of my particular art, my most personal work and creation, to turn to.”
— Wagner’s address at the ceremony for the laying of the foundation stone of the Festival Theater in Bayreuth, May 22 (Wagner’s birthday), 1872
“The new computer package DISCRETA which was created in Bayreuth is in the process of permanent development.”
— “A Computer Approach to the Enumeration of Block Designs Which Are Invariant With Respect to a Prescribed Permutation Group”
The above is a preprint from Dresden.
See, too, the work of Bierbrauer, who received his doctorate at Mainz in 1977 and taught at Heidelberg from 1977 to 1994. Bierbrauer’s lecture notes give a particularly good background for the concepts involved in my Diamond Theory, in the tradition of Witt and Artin. See
Introduction to Group Theory
and Applications,
by Jürgen Bierbrauer, 138 pp., PostScript
Comments Off on Thursday May 22, 2003
Friday, April 25, 2003
Mark
Today is the feast of Saint Mark. It seems an appropriate day to thank Dr. Gerald McDaniel for his online cultural calendar, which is invaluable for suggesting blog topics.
Yesterday's entry "Cross-Referenced" referred to a bizarre meditation of mine titled "The Matthias Defense," which combines some thoughts of Nabokov on lunacy with some of my own thoughts on the Judeo-Christian tradition (i.e., also on lunacy). In this connection, the following is of interest:
From a site titled Meaning of the Twentieth Century —
"Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called 'Innovation in Physics,' he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: 'We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.' To that Freeman added: 'When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!' "
— Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147
It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson. It is no less crazy than Christianity, and has a certain mad simplicity that perhaps improves on some of that religion's lunatic doctrines.
Some fruits of the "162 theory" —
Searching on Google for muses 162, we find the following Orphic Hymn to Apollo and a footnote of interest:
27 Tis thine all Nature's music to inspire,
28 With various-sounding, harmonising lyre;
29 Now the last string thou tun'ft to sweet accord,
30 Divinely warbling now the highest chord….
"Page 162 Verse 29…. Now the last string…. Gesner well observes, in his notes to this Hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras, to Plato. Now, according to the Orphic and Pythagoric doctrine, the lyre of Apollo is an image of the celestial harmony…."
For the "highest chord" in a metaphorical sense, see selection 162 of the 1919 edition of The Oxford Book of English Verse (whose editor apparently had a strong religious belief in the Muses (led by Apollo)). This selection contains the phrase "an ever-fixèd mark" — appropriately enough for this saint's day. The word "mark," in turn, suggests a Google search for the phrase "runes to grave" Hardy, after a poem quoted in G. H. Hardy's A Mathematician's Apology.
Such a search yields a website that quotes Housman as the source of the "runes" phrase, and a further search yields what is apparently the entire poem:
Smooth Between Sea and Land
by A. E. Housman
Smooth between sea and land
Is laid the yellow sand,
And here through summer days
The seed of Adam plays.
Here the child comes to found
His unremaining mound,
And the grown lad to score
Two names upon the shore.
Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?
Tell me of runes to grave
That hold the bursting wave,
Or bastions to design
For longer date than mine.
Shall it be Troy or Rome
I fence against the foam
Or my own name, to stay
When I depart for aye?
Nothing: too near at hand
Planing the figured sand,
Effacing clean and fast
Cities not built to last
And charms devised in vain,
Pours the confounding main.
(Said to be from More Poems (Knopf, 1936), p. 64)
Housman asks the reader to tell him of runes to grave or bastions to design. Here, as examples, are one rune and one bastion.
Dagaz: (Pronounced thaw-gauze, but with the "th" voiced as in "the," not unvoiced as in "thick") (Day or dawn.)
From Rune Meanings:
Dagaz means "breakthrough, awakening, awareness. Daylight clarity as opposed to nighttime uncertainty. A time to plan or embark upon an enterprise. The power of change directed by your own will, transformation. Hope/happiness, the ideal. Security and certainty. Growth and release. Balance point, the place where opposites meet."
Also known as "the rune of transformation."
For the Dagaz rune in another context, see Geometry of the I Ching. The geometry discussed there does, in a sense, "hold the bursting wave," through its connection with Walsh functions, hence with harmonic analysis.
Temple of Athena Nike on the Nike Bastion, the Acropolis, Athens. Here is a relevant passage from Paul Valéry's Eupalinos ou L'Architecte about another temple of four columns:
Et puis… Écoute, Phèdre (me disait-il encore), ce petit temple que j'ai bâti pour Hermès, à quelques pas d'ici, si tu savais ce qu'il est pour moi ! — Où le passant ne voit qu'une élégante chapelle, — c'est peu de chose: quatre colonnes, un style très simple, — j'ai mis le souvenir d'un clair jour de ma vie. Ô douce métamorphose ! Ce temple délicat, nul ne le sait, est l'image mathématique d'une fille de Corinthe que j'ai heureusement aimée. Il en reproduit fidèlement les proportions particulières. Il vit pour moi !
Four columns, in a sense more suited to Hardy's interests, are also a recurrent theme in The Diamond 16 Puzzle and Diamond Theory.
Apart from the word "mark" in The Oxford Book of English Verse, as noted above, neither the rune nor the bastion discussed has any apparent connection with the number 162… but seek and ye shall find.
Comments Off on Friday April 25, 2003
Sunday, April 13, 2003
Palm Sunday, Part II:
Cold Mountain
From the notes to the CD of Songs From the Mountain (John Herrmann, Dirk Powell, Tim O’Brien):
“John [Herrmann, banjo player] would like to dedicate his work on this recording to Philip Kapleau Roshi, Kalu Rimpoche, and Harada Tangen Roshi, who all know the way to Cold Mountain….”
See Buddha’s Birthday (April 8) and The Diamond Project.
“What are you thinking of? What thinking? What?
I never know what you are thinking. Think.”
— Tom Eliot, The Waste Land
“I am thinking…
… of the midnight picnic
Once upon a time….”
— Suzanne Vega, “Tom’s Diner“
“De donde crece la palma” — Song lyric
From On Beauty, by Elaine Scarry, Princeton University Press, 1999, a quotation from Homer —
“in Delos, beside Apollo’s altar —
the young slip of a palm-tree
springing into the light.”
See also A Mass for Lucero and The Shining of Lucero.
“How much story do you want?”
— George Balanchine
Comments Off on Sunday April 13, 2003
Sunday, March 16, 2003
Letter
On this date in 1850, Nathaniel Hawthorne’s novel
“The Scarlet Letter” was first published.
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“Ye see how large a letter
I have written….”
— St. Paul, Galatians
Chapter 6, Verse 11
Einstein and Oppenheimer
at the Institute for
Advanced Study, Princeton
Einstein wrote a rather
famous letter to FDR;
Oppenheimer is known as
the father of the A-bomb.
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Song of God
From the Hindu holy scripture Bhagavad Gita (Song of God):
“Of orators, I am the speech; of letters the first one, A; I am imperishable time; the Creator whose face is everywhere; death that devours all things.”
— Bhagavad Gita 10.32-33,
tr. by Stephen Mitchell,
Harmony Books, 2000
“…  “
— Suzan-Lori Parks,
playwright,
in today’s New York Times
Suzan-Lori Parks
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See also my note of two years ago,
“Random Thoughts for St. Patrick’s Eve.”
For more on Oppenheimer and the Bhagavad Gita, see
“Fat Man and Dancing Girl.”
Comments Off on Sunday March 16, 2003
Thursday, March 13, 2003
ART WARS:
From The New Yorker, issue of March 17, 2003, Clive James on Aldous Huxley:
“The Perennial Philosophy, his 1945 book compounding all the positive thoughts of West and East into a tutti-frutti of moral uplift, was the equivalent, in its day, of It Takes a Village: there was nothing in it to object to, and that, of course, was the objection.”
For a cultural artifact that is less questionably perennial, see Huxley’s story “Young Archimedes.”
Plato, Pythagoras, and
the diamond figure
Plato’s Diamond in the Meno
Plato as a precursor of Gerard Manley Hopkins’s “immortal diamond.” An illustration shows the ur-diamond figure.
Plato’s Diamond Revisited
Ivars Peterson’s Nov. 27, 2000 column “Square of the Hypotenuse” which discusses the diamond figure as used by Pythagoras (perhaps) and Plato. Other references to the use of Plato’s diamond in the proof of the Pythagorean theorem:
Huxley:
“… and he proceeded to prove the theorem of Pythagoras — not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself….
‘You see,’ he said, ‘it seemed to me so beautiful….’
I nodded. ‘Yes, it’s very beautiful,’ I said — ‘it’s very beautiful indeed.'”
— Aldous Huxley, “Young Archimedes,” in Collected Short Stories, Harper, 1957, pp. 246 – 247
Heath:
Sir Thomas L. Heath, in his commentary on Euclid I.47, asks how Pythagoreans discovered the Pythagorean theorem and the irrationality of the diagonal of a unit square. His answer? Plato’s diamond.
(See Heath, Sir Thomas Little (1861-1940),
The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publications, New York, 1956.
Other sites on the alleged
“diamond” proof of Pythagoras
Colorful diagrams at Cut-the-Knot
Illustrated legend of the diamond proof
Babylonian version of the diamond proof
For further details of Huxley’s story, see
The Practice of Mathematics,
Part I, by Robert P. Langlands, from a lecture series at the Institute for Advanced Study, Princeton.
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From the New Yorker Contributors page for St. Patrick’s Day, 2003:
“Clive James (Books, p. 143) has a new collection, As of This Writing: The Essential Essays, 1968-2002, which will be published in June.”
See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.
Comments Off on Thursday March 13, 2003
Tuesday, March 11, 2003
ART WARS:
The Producers
Simon and Garfunkel's Tribute to Synchronicity:
Fools, said I,
you do not know
Silence like
a cancer grows.
Hear my words
that I might teach you.
Take my arms
that I might reach you….
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Dummköpfe, sagte ich,
ihr wißt nicht,
daß die Stille wie
ein Krebs wächst.
Hört meine Worte,
die ich euch sage.
Nehmt meine Hände,
die ich euch reiche….
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And the people
bowed and prayed
To the neon god
they made.
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Und die Menschen
verbeugten sich vor dem
Neon-Gott, den sie schufen,
und beteten zu ihm.
— Paul Simon
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For more on Jung, see
See also the Synchronicity album of The Police,
inducted last night into the Rock and Roll Hall of Fame.
Comments Off on Tuesday March 11, 2003
Tuesday, March 4, 2003
Fearful Symmetry
I just Googled this phrase and found the following site, which turns out to be related to my previous entry on the Bead Game and the death of John P. Thompson.
Fearful Symmetry:
The Music Master’s Lecture,
by Daniel d’Quincy.
This in turn links to an excerpt from The Glass Bead Game that includes this passage:
“I suddenly realized that in the language, or at any rate in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created.”
It is very easy to get dangerously confused about holiness, but here are some relevant quotes:
“You will have to allow me to digress a bit in order to bring ourselves to a sufficiently elevated perspective… I warn you, it will require an attitude of playfulness on your part. Our approach will aim more at sincerity than seriousness. The attitude I’m aiming at is best expressed, I suppose, in the playing of a unique game, known by its German name as Das Glasperlenspiel, and which we may translate as the Glass Bead Game.”
— Daniel d’Quincy, Fearful Symmetry
“7:11”
— God himself said this, at least according to the previous entry and to my Jan. 28 entry, State of the Communion.
“Seven is heaven.”
— See my web page Eight is a Gate.
“An excellent example of a ‘universal’ in the sense of Charles Williams, Jung, or Plato is Hexagram 11 in China’s 3,000-year-old classic, the I Ching:
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‘Heaven and earth unite:
the image of PEACE.’
(Wilhelm/Baynes translation,
Princeton University Press, 1967)”
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— S. H. Cullinane, Plato, Pegasus, and the Evening Star
Thus we may associate the numbers 7 and 11 with the notions of heaven and peace; for a somewhat darker association of the time 7:11 with Kali as Time the Destroyer, see my last entry and also my previous entries
Fat Man and Dancing Girl (Feb. 18, 2003), and
Time and Eternity (Feb. 1, 2003).
Monday, February 24, 2003
Moulins Rouges
Today is the birthday of composer Michel Legrand (“The Windmills of Your Mind”) and of philologist Wilhelm Grimm (Grimms’ Fairy Tales).
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Red
Windmill
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Red
Mill
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Rode
Molen
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See the following past entries:
October 6, 2002: “Twenty-first Century Fox”
November 7, 2002: “Endgame”
November 8, 2002: “Religious Symbolism at Princeton”
January 5, 2003: “Whirligig”
January 5, 2003: “Culinary Theology”
January 6, 2003: “Dead Poet in the City of Angels”
January 31, 2003: “Irish Fourplay”
February 1, 2003; “Time and Eternity”
February 5, 2003: “Release Date”
Comments Off on Monday February 24, 2003
Tuesday, February 18, 2003
Fat Man and Dancing Girl
Physicist J. Robert Oppenheimer, portrayed in the film "Fat Man and Little Boy," died on this date in 1967.
He is sometimes called the "father of the A-bomb." He said that at the time of the first nuclear test he thought of a line from the Sanskrit holy book, the Bhagavad Gita: "I am become death, the destroyer of worlds." The following gives more details.
The Bomb of the Blue God
M. V. Ramana
Center for Energy and Environmental Studies, Princeton University
Published in SAMAR: South Asian Magazine for Action and Reflection, Issue 13
Oppenheimer had learned Sanskrit at Berkeley so as to read the Gita in the original; he always kept a worn pink copy on the bookshelf closest to his desk. It is therefore likely that he may have actually thought of the original, Sanskrit, verse rather than the English translation. The closest that fits this meaning is in the 32nd verse from the 11th chapter of the Gita.
kalosmi lokaksaya krt pravrddho
This literally means: I am kAla, the great destroyer of Worlds. What is intriguing about this verse, then, is the interpretation of kAla by Jungk and others to mean death. While death is technically one of the meanings of kAla, a more common one is time. Indeed, the translations of the Gita by S. Radhakrishnan, A. C. Bhaktivedanta, Nataraja Guru and Eliot Deutsch say precisely that. One exception to this, however, is the 1929 translation by Arthur Ryder. And, indeed, in a 1933 letter to his brother, Robert Oppenheimer does mention that he has "been reading the Bhagavad Gita with Ryder and two other Sanskritists." The misinterpretation, therefore, may not have been the fault of Oppenheimer or Jungk. Nevertheless, the verse does not have anything to do with an apocalyptic or catastrophic destruction, as most people have interpreted it in connection with nuclear weapons. When kAla is understood as time, the meaning is drastically changed to being a reminder of our mortality and finite lifetimes as also the lifetimes of everything else in this world (including plutonium and uranium, despite their long, long, half-lives!). It then becomes more akin to western notions of the "slow march of time" and thus having little to do with the immense destruction caused by a nuclear explosion. While the very first images that arose in the father of the atomic bomb are a somewhat wrong application of Hindu mythology, his recollection of the Bhagavad Gita may have been quite pertinent. As is well known, the Bhagavad Gita was supposedly intended to persuade Arjuna to participate in the Kurukshetra battle that resulted in the killing of thousands. Thus, Oppenheimer may well have been trying to rationalize his involvement in the development of a terrible weapon.
Source: Google cache of
http://www.geocities.com/CollegePark/5409/samar_bluegod.pdf
See also
http://www.samarmagazine.org/archive/article.php?id=36.
"KAla" (in the Harvard-Kyoto transliteration scheme) is more familiar to the West in the related form of Kali, a goddess sometimes depicted as a dancing girl; Kali is related to kAla, time, according to one website, as "the force which governs and stops time." See also the novel The Fermata, by Nicholson Baker.
The fact that Oppenheimer thought of Chapter 11, verse 32, of the Gita may, as a mnemonic device, be associated with the use of the number 1132 in Finnegans Wake.
See 1132 A. D. & Saint Brighid, and my weblog entries of January 5 (Twelfth Night and the whirligig of time), January 31 (St. Bridget's Eve), and February 1 (St. Bridget's Day), 2003.
Friday, February 14, 2003
Movie Date
For John and Klara von Neumann,
because they gave good parties:
"We gotta hurry or it's gonna be dark
'fore we get home!"
Home Before Dark
A song by Judy Collins:
I won't be long
Don't worry about me
I'll be home before dark
Plato's Cave Valentine's Day Schedule
at UA Market Fair Movies,
3521 Route 1
Princeton, NJ 08540
Before-dark* showtime:
The Pianist, 5:30 PM
After-dark† showtime:
The Recruit, 7:15 PM
* "Swiftly flow the years."
† "Nothing is as it seems."
Thursday, February 13, 2003
From Plato’s Cave
(Von Neumann’s Song, Part III)
In this entry we return to the classic words of the Hollywood Argyles as they sing a paean of praise to St. John von Neumann:
He’s the king of the jungle jive.
Look at that caveman go!
This meditation is prompted by a description of caveman life by the functional analysis working group at the University of Tübingen:
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John von
Neumann
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“Soon Freud, soon mourning,
Soon Fried, soon fight.
Nevertheless who know this language?”
(Language courtesy of
Google’s translation software)
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Picture of von Neumann courtesy of
Princeton University Library
Comments Off on Thursday February 13, 2003
Wednesday, February 5, 2003
Feast of Saint Marianne
On this date in 1972, poet and Presbyterian saint Marianne Moore died in New York City.
For why she was a saint, see the excellent article by Samuel Terrien,
“Marianne Moore: Poet of Secular Holiness,”
from Theology Today, Vol. 47, No. 4, January 1991, published by Princeton Theological Seminary, Princeton, N. J.
Terrien quotes the following Moore poem:
Tonight’s site music, though not played by Gieseking himself, is, in honor of Moore, the following work by Scarlatti from the Classical Music Archives:
Scarlatti’s Sonata in E major, andante comodo (Longo 23 = Kirkpatrick 380 = Pestelli 483)
To purchase a recording of Gieseking playing this work,
click here.
Comments Off on Wednesday February 5, 2003
Friday, January 10, 2003
Story
"How much story do you want?"
— George Balanchine
While researching yesterday's entry on Balanchine, Apollo, and the nine Muses, I came across this architect's remarks, partially quoted yesterday and continued here:
"The icon that I use for this element is the nine-fold square…. This is the garden of Apollo, the field of Reason…. This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale, as described to me by Professor Hersey."
— Duncanology Part 3
Checking this out yesterday, I came across the following at a Yale University Art Gallery site:
"This exhibition of nine boldly colored, asymmetrically designed quilts selected from a private collection will be displayed in the Matrix Gallery….
With the guidance of Professor Maude Southwell Wahlman, author of 'Signs and Symbols: African Images in African American Quilts,' the collector has explored and gathered examples…."
Exploring and gathering examples myself today, I received a book in the mail — W. M. Spackman's On the Decay of Humanism (Rutgers University Press, 1967) — and picked up a second-hand book at a sale — Barbara Michaels's Stitches in Time (Harper Collins Publishers, 1995).
The Spackman book includes the following poem at the end:
In sandarac etui for sepulchre
lies the cered body of a poisoned queen;
and in her mouth and hair, and at her feet,
and in the grey folds of her winding-sheet,
there sifts a dreamy powder, smooth and green,
the magic of an idle sorcerer,
an ancient spell, cast when the shroud was spun.
In death her hands clasp amourously a bowl
that still contains the fragments of her soul,
a tale of Beauty sought, and Beauty won,
his false lips kissed, and Beauty dead for her.
— Alexander B. Griswold, Princeton '28, in the
Nassau Literary Magazine of December 1925
From a synopsis of Michaels's Stitches in Time:
"Michaels follows Rachel, a graduate student studying women's crafts–weaving, spinning, quilting, embroidery–and the superstitions connected with them. Linking all important rites of passage to the garments created as markers of these occasions leads Rachel to her theory: in societies in which magic was practiced, the garment was meant to protect its wearer. She gains evidence that her theory is valid when an evil antique bridal quilt enters her life."
Although Stitches in Time is about a quilt — stitched, not spun — Griswold's line
"an ancient spell, cast when the shroud was spun"
is very closely related to the evil spell in Michaels's book.
The above events display a certain synchronicity that Wallace Stevens might appreciate, especially in light of the following remark in a review of Stitches in Time:
"…the premise is too outlandish for even the suspension of disbelief…." (Publishers Weekly, 4/24/95)
Stevens might reply,
The very man despising honest quilts
Lies quilted to his poll in his despite.
— "The Comedian as the Letter C," Part V
Finally, those who prefer stories to the more formal qualities of pure dance (ballet) pure mathematics (see previous entry), pure (instrumental) music, and pure (abstract, as in quilt designs) art, can consult the oeuvre of Jodie Foster — as in my
Pearl Harbor Day entry on Buddhism.
An art historian named Griswold — perhaps that very same Griswold quoted above — might have a thing or two to say to Jodie on her recent film "Anna and the King." In the April, 1957, issue of The Journal of the Siam Society, Alexander B. Griswold takes issue with Broadway's and Hollywood's "grotesque caricature" of Siamese society, and ultimately with Anna herself:
"The real fault lies in the two books they ultimately spring from — The English Governess at the Court of Siam and The Romance of the Harem — both written by Mrs. Anna Leonowens.''
Is a puzzlement.
See also The Diamond 16 Puzzle for some quilt designs.
Comments Off on Friday January 10, 2003
Wednesday, December 4, 2002
Symmetry and a Trinity
From a web page titled Spectra:
"What we learn from our whole discussion and what has indeed become 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. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms . . ."
— Hermann Weyl in Symmetry, Princeton University Press, 1952, page 144
"… any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated . . .
Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:
X = aA + bB + cC.
Now suppose another color Y is made from the same three colors:
Y = a'A + b'B + c'C.
Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:
Z = X + Y = (a + a')A + (b + b')B + (c + c')C.
It is just like the mathematics of the addition of vectors, where (a, b, c ) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians."
— According to the author of the Spectra site, this is Richard Feynman in Elementary Particles and the Laws of Physics, The 1986 Dirac Memorial Lectures, by Feynman and Steven Weinberg, Cambridge University Press, 1989.
These two concepts — symmetry as invariance under a group of transformations, and complicated things as linear combinations (the technical name for Feynman's sums) of simpler things — underlie much of modern mathematics, both pure and applied.
Comments Off on Wednesday December 4, 2002
Tuesday, December 3, 2002
Symmetry, Invariance, and Objectivity
The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.
On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:
"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."
Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):
"Objectivity means invariance with respect to the group of automorphisms."
It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.
I pointed this out on June 10, 2002.
For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).
Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.
Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.
A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:
"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged." (p. 69)
This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."
Monday, November 25, 2002
The Artist’s Signature
This title is taken from the final chapter of Carl Sagan’s novel Contact.
“There might be a game in which paper figures were put together to form a story, or at any rate were somehow assembled. The materials might be collected and stored in a scrap-book, full of pictures and anecdotes. The child might then take various bits from the scrap-book to put into the construction; and he might take a considerable picture because it had something in it which he wanted and he might just include the rest because it was there.”
— Ludwig Wittgenstein, Lectures and Conversations on Aesthetics, Psychology and Religious Belief
“Not games. Puzzles. Big difference. That’s a whole other matter. All art — symphonies, architecture, novels — it’s all puzzles. The fitting together of notes, the fitting together of words have by their very nature a puzzle aspect. It’s the creation of form out of chaos. And I believe in form.”
— Stephen Sondheim, in Stephen Schiff, Deconstructing Sondheim,” The New Yorker, March 8, 1993, p. 76
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“All goods in this world, all beauties, all truths, are diverse and partial aspects of one unique good. Therefore they are goods which need to be ranged in order. Puzzle games are an image of this operation. Taken all together, viewed from the right point and rightly related, they make an architecture. Through this architecture the unique good, which cannot be grasped, becomes apprehensible. All architecture is a symbol of this, an image of this. The entire universe is nothing but a great metaphor.”
— Simone Weil, sister of Princeton mathematician André Weil, First and Last Notebooks, p. 98
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This passage from Weil is quoted in
Gateway to God,
p. 42, paperback, fourth impression,
printed in Glasgow in 1982 by
Fontana Books
“He would leave enigmatic messages on blackboards,
signed Ya Ya Fontana.”
— Brian Hayes on John Nash,
The Sciences magazine, Sept.-Oct., 1998
“I have a friend who is a Chief of the Aniunkwia (Cherokee) people and I asked him the name of the Creator in which
he replied… Ya Ho Wah. This is also how it is spoken in Hebrew. In my native language it is spoken
Ya Ya*,
which is also what Moses was told
at the ‘Burning Bush’ incident.”
— “Tank” (of Taino ancestry), Bronx, NY, Wednesday, April 17, 2002
From a website reviewing books published by
Fontana:
1/17/02: NEW YORK (Variety) – Russell Crowe is negotiating to star in 20th Century Fox’s “Master and Commander,” the Peter Weir-directed adaptation of the Patrick O’Brian book series.
Hmmm.
*For another religious interpretation of this phrase, see my note of October 4, 2002, “The Agony and the Ya-Ya.”
Comments Off on Monday November 25, 2002
Saturday, November 9, 2002
Birthdate of Hermann Weyl
Result of a Google search.
Category: Science > Math > Algebra > Group Theory
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Weyl, H.: Symmetry.
Description of the book Symmetry by Weyl, H., published by Princeton University Press. … pup.princeton.edu/titles/
865.html – 7k – Nov. 8, 2002
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Sponsored Link
Symmetry Puzzle
New free online puzzle illustrates
the mathematics of symmetry.
m759.freeservers.com/puzzle.
html
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Quotation from Weyl’s Symmetry:
“Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.”
In honor of Princeton University, of Sylvia Nasar (see entries of Nov, 6), of the Presbyterian Church (see entry of Nov. 8), and of Professor Weyl (whose work partly inspired the website Diamond Theory), this site’s background music is now Pink Floyd’s
Updates of Friday, November 15, 2002:
In order to clarify the meaning of “Shine” and “Crazy” in the above, consult the following —
To accompany this detailed exegesis of Pink Floyd, click here for a reading by Marlon Brando.
For a related educational experience, see pages 126-127 of The Book of Sequels, by Henry Beard, Christopher Cerf, Sarah Durkee, and Sean Kelly (Random House paperback, 1990).
Speaking of sequels, be on the lookout for Annie Dillard’s sequel to Teaching a Stone to Talk, titled Teaching a Brick to Sing.
Tuesday, September 24, 2002
The Shining of Lucero
From my journal note, “Shining Forth“:
The Spanish for “Bright Star” is “Lucero.”
The Eye of the Beholder:
When you stand in the dark and look at a star a hundred light years away, not only have the retarded light waves from the star been travelling for a hundred years toward your eyes, but also advanced waves from your eyes have reached a hundred years into the past to encourage the star to shine in your direction.
— John Cramer, “The Quantum Handshake“
From Broken Symmetries, by Paul Preuss, 1983:
He’d toyed with “psi” himself…. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand — for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox….
Peter [Slater] had never thirsted after “hidden variables” to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods.
………………
Those so-called crazy psychics were too sane, that was their problem — they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry. (Ch. 16)
From Secret Passages, by Paul Preuss, 1997:
Minakis caught up and walked beside him in silence, moving with easy strides over the bare ground, listening as Peter [Slater] spoke. “Delos One was ten years ago — quantum theory seemed as natural as water to me then; I could play in it without a care. If I’d had any sense of history, I would have recognized that I’d swallowed the Copenhagen interpretation whole.”
“Back then, you insisted that the quantum world is not a world at all,” Minakis prompted him. “No microworld, only mathematical descriptions.”
“Yes, I was adamant. Those who protested were naive — one has to be willing to tolerate ambiguity, even to be crazy.”
“Bohr’s words?”
“The party line. Of course Bohr did say, ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’ Meaning that when we start to talk what sounds like philosophy, our colleagues should rip us to pieces.” Peter smiled. “They smell my blood already.”
……………… Peter glanced at Minakis. “Let’s say there are indications — I have personal indications — not convincing, perhaps, but suggestive, that the quantum world penetrates the classical world deeply.” He was silent for a moment, then waved his hand at the ruins. “The world of classical physics, I mean. I suppose I’ve come to realize that the world is more than a laboratory.”
“We are standing where Apollo was born,” Minakis said. “Leto squatted just there, holding fast to a palm tree, and after nine days of labor gave birth to the god of light and music….”
To Lucero, in memory of
1962 in Cuernavaca
From On Beauty, by Elaine Scarry,
Princeton University Press, 1999 —
“Homer sings of the beauty of particular things. Odysseus, washed up on shore, covered with brine, having nearly drowned, comes upon a human community and one person in particular, Nausicaa, whose beauty simply astonishes him. He has never anywhere seen a face so lovely; he has never anywhere seen any thing so lovely….
I have never laid eyes on anyone like you,
neither man nor woman…
I look at you and a sense of wonder takes me.
Wait, once I saw the like —
in Delos, beside Apollo’s altar —
the young slip of a palm-tree
springing into the light.”
From Secret Passages, by Paul Preuss, 1997:
“When we try to look inside atoms,” Peter said, “not only can we not see what’s going on, we cannot even construct a coherent picture of what’s going on.”
“If you will forgive me, Peter,” Minakis said, turning to the others. “He means that we can construct several pictures — that light and matter are waves, for example, or that light and matter are particles — but that all these pictures are inadequate. What’s left to us is the bare mathematics of quantum theory.”
…. “Whatever the really real world is like, my friend, it is not what you might imagine.”
………………
Talking physics, Peter tended to bluntness. “Tell me more about this real world you imagine but can’t describe.”
Minakis turned away from the view of the sunset. “Are you familiar with John Cramer’s transactional interpretation of quantum mechanics?”
“No I’m not.”
………………
“Read Cramer. I’ll give you his papers. Then we can talk.”
From John Cramer, “The Quantum Handshake“:
Advanced waves could perhaps, under the right circumstances, lead to “ansible-type” FTL communication favored by Le Guin and Card….
For more on Le Guin and Card, see my journal notes below.
For more on the meaning of “lucero,” see the Wallace Stevens poem “Martial Cadenza.”
Comments Off on Tuesday September 24, 2002
Thursday, September 19, 2002
Fermat’s Sombrero
Mexican singer Vincente Fernandez holds up the Latin Grammy award (L) for Best Ranchero Album he won for “Mas Con El Numero Uno” and the Latin Grammy Legend award at the third annual Latin Grammy Awards September 18, 2002 in Hollywood. REUTERS/Adrees Latif
From a (paper) journal note of January 5, 2002:
Princeton Alumni Weekly
January 24, 2001
The Sound of Math:
Turning a mathematical theorem
and proof into a musical
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How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor?
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Wallace Stevens:
Poet of the American Imagination
Consider these lines from
“Six Significant Landscapes” part VI:
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses-
As, for example, the ellipse of the half-moon-
Rationalists would wear sombreros.
Addendum of 9/19/02: See also footnote 25 in
Theological Method and Imagination
by Julian N. Hartt
Comments Off on Thursday September 19, 2002
Wednesday, August 28, 2002
Requiem for a Critic
A 1999 Mike Melillo Trio album, “Bopcentric,” includes the above compositions.
“Melillo is a striking pianist, a chameleon who has a Bud Powell touch on a Charlie Parker be bop number, who evokes appropriate echoes of Thelonious Monk”.
— John S. Wilson in The New York Times
From sleeve notes by Orrin Keepnews at
The Thelonious Monk Website:
For many years regarded as an awesome genius, but one whose ideas were too far-out for general consumption, Monk now seems finally to be gaining long-deserved acceptance…. some critics feel that he is becoming (as John S. Wilson has put it) “increasingly lucid.”
From The New York Times of August 28, 2002:
John S. Wilson, the first critic to write regularly about jazz and popular music in The New York Times, died yesterday at a nursing home in Princeton, N.J. He was 89 and lived in Princeton.
Comments Off on Wednesday August 28, 2002
Tuesday, August 27, 2002
The Hero and the Dark Lady
From a Fictionwise eBooks summary of Mike Resnick's novel The Dark Lady…
Leonardo, an art historian of the far future, is given a mission…
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"His instructions: Search the galaxy for any piece of art bearing the image of his obsession: the mysterious Dark Lady, a beautiful and somber human female whose exact likeness, he has secretly discovered, appears in paintings and sculptures throughout history–dating all the way back to Earth's ancient Rome.
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Leonardo's research reveals the link between the artists of the Dark Lady: human men who voluntarily risk their lives. If she appears to men who court death, she may be their Angel of Death … or, as Leonardo hopes, the female of an ancient… legend–The Mother of All Things."
Today, August 27 (or tomorrow, according to some accounts), is the date of death of a great actor, Robert Shaw, who died at 51 in 1978. If in real life he was anything like the brave men he played… King Henry VIII, SPECTRE assassin Red Grant, Panzer commander Colonel Hessler, and shark hunter Captain Quint… he, if anyone, deserved to be greeted in heaven by the Dark Lady.
For a more scholarly treatment of the Dark Lady, see this Princeton University Press site.
Comments Off on Tuesday August 27, 2002
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