Log24

Thursday, July 5, 2007

Thursday July 5, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:11 pm

In Defense of
Plato’s Realism

(vs. sophists’ nominalism–
see recent entries.)

Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:

The Eightfold Cube and its Inner Structure

Click on image for details.

As noted in an entry,
Plato, Pegasus, and
the Evening Star,
linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:

Trigram of K'un, the Receptive

K’un
The Receptive

_____________________________________________

Update of Nov. 30, 2013:

From  a little-known website in Kuala Lumpur:
(Click to enlarge.)

The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: , — m759 @ 4:30 pm

Schopenhauer on the Kernel of Eternity

Philos Website —

"Ich aber, hier auf dem objektiven Wege, bin jetzt bemüht, das Positive der Sache nachzuweisen, daß nämlich das Ding an sich von der Zeit und Dem, was nur durch sie möglich ist, dem Entstehen und Vergehen, unberührt bleibt, und daß die Erscheinungen in der Zeit sogar jenes rastlos flüchtige, dem Nichts zunächst stehende Dasein nicht haben könnten, wenn nicht in ihnen ein Kern aus der Ewigkeit* wäre. Die Ewigkeit ist freilich ein Begriff, dem keine Anschauung zum Grunde liegt: er ist auch deshalb bloß negativen Inhalts, besagt nämlich ein zeitloses Dasein. Die Zeit ist demnach ein bloßes Bild der Ewigkeit, ho chronos eikôn tou aiônos,** wie es Plotinus*** hat: und ebenso ist unser zeitliches Dasein das bloße Bild unsers Wesens an sich. Dieses muß in der Ewigkeit liegen, eben weil die Zeit nur die Form unsers Erkennens ist: vermöge dieser allein aber erkennen wir unser und aller Dinge Wesen als vergänglich, endlich und der Vernichtung anheimgefallen."

*    "a kernel of eternity"
**  "Time is the image of eternity."
*** "wie es Plotinus hat"–
       Actually, not Plotinus, but Plato,
       according to Diogenes Laertius.

Related material:

Time Fold,

J. N. Darby,
"On the Greek Words for
Eternity and Eternal
(aion and aionios),"

Carl Gustav Jung, Aion,
which contains the following
four-diamond figure,

Jung's four-diamond figure

and Jung and the Imago Dei.

Tuesday, May 22, 2007

Tuesday May 22, 2007

Filed under: General,Geometry — m759 @ 7:11 am
 
Jewel in the Crown

A fanciful Crown of Geometry

The Crown of Geometry
(according to Logothetti
in a 1980 article)

The crown jewels are the
Platonic solids, with the
icosahedron at the top.

Related material:

"[The applet] Syntheme illustrates ways of partitioning the 12 vertices of an icosahedron into 3 sets of 4, so that each set forms the corners of a rectangle in the Golden Ratio. Each such rectangle is known as a duad. The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges, so there are 15 duads.

Each partition of the vertices into duads is known as a syntheme. There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while the other 10 consist of duads that share a common line of intersection."

— Greg Egan, Syntheme

Duads and synthemes
(discovered by Sylvester)
also appear in this note
from May 26, 1986
(click to enlarge):

 

Duads and Synthemes in finite geometry

The above note shows
duads and synthemes related
to the diamond theorem.

See also John Baez's essay
"Some Thoughts on the Number 6."
That essay was written 15 years
ago today– which happens
to be the birthday of
Sir Laurence Olivier, who,
were he alive today, would
be 100 years old.

Olivier as Dr. Christian Szell

The icosahedron (a source of duads and synthemes)

"Is it safe?"

Sunday, April 22, 2007

Sunday April 22, 2007

Filed under: General,Geometry — Tags: — m759 @ 8:31 pm
Built
continued from
March 25, 2006

In honor of Scarlett Johansson's recent London films "Match Point" and "Scoop," here is a link to an entry of Women's History Month, 2006, with a discussion of an exhibition of the works of artist Liza Lou at London's White Cube Gallery.  That entry includes the following illustrations:


The image “http://www.log24.com/log/pix06/060325-WhiteCube.jpg” cannot be displayed, because it contains errors.

 

This work might aptly be
  retitled "Brick Shithouse."

Related material:

The artist's self-portrait

The image “http://www.log24.com/log/pix06/060325-LizaLouSelfPortrait.jpg” cannot be displayed, because it contains errors.

See also this morning's entry —

"She's a brick… house…
The lady's stacked
   and that's a fact,
Ain't holdin' nothin' back."

— and last year's entry
on this date:

The image “http://www.log24.com/log/pix06/060422-Johansson1.jpg” cannot be displayed, because it contains errors.

"Her wall is filled with pictures,
She gets 'em one by one."

The bricks and "white cube"
above and in this morning's entry
may be contrasted with the
bricks of Diamonds and Whirls
and the cube of On Beauty.

  Poetic allusions such as these
may help provide
entertainment in the afterlife
for Beavis, Butt-Head, and
other inmates of Plato's Cave:

The image “http://www.log24.com/log/pix07/070328-PlatoCave.jpg” cannot be displayed, because it contains errors.

"The Garden of Eden is behind us
and there is no road back to innocence;
we can only go forward."

— Anne Morrow Lindbergh,
Earth Shine, p. xii

 

Tuesday, January 23, 2007

Tuesday January 23, 2007

Filed under: General — m759 @ 12:00 pm
Quine vs. Kierkegaard

“The most prominent critic
of the modal notions is Quine.
Throughout his career, he has
argued against the use of notions
like necessity and possibility.”

— Michael J. Loux,
Note 1 of Chapter 5,
“The Necessary and the Possible,”
in Metaphysics:
A Contemporary Introduction
(Routledge, second edition,
January 1, 2002)

“Personality is a synthesis of
possibility and necessity.”

— Soren Kierkegaard,
The Sickness Unto Death

Related material:

Plato, Pegasus,
and the Evening Star

Diamonds Are Forever

Dream a Little Dream

Update of 3:45 PM:

From Arts & Letters Daily
this afternoon–

“Existentialism is not all gloom,
even if Heidegger looks pretty sour
in those photos. It’s a philosophy
that America needs now, says
the late Robert Solomon…
more … obit”

See also Jan. 2,
the date of
Solomon’s death
in Switzerland,
and click on the
following symbol
from that date:

The image “http://www.log24.com/log/pix07/070103-DoubleCross.jpg” cannot be displayed, because it contains errors.

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General,Geometry — m759 @ 9:00 pm
Logos and Logic
(private, cut from prev. entry)

The diamond is used in modal logic to symbolize possibility.

  The 3×3 grid may also be used
to illustrate “possibility.”  It leads,
as noted at finitegeometry.org, to
the famed “24-cell,” which may be
pictured either as the diamond
figure from Plato’s Meno —

The image “http://www.log24.com/theory/images/poly-24cell-sm.jpg” cannot be displayed, because it contains errors.

Click for details.

  — or as a figure
with 24 vertices:

The image “http://www.log24.com/theory/images/poly-24cell-02sm.jpg” cannot be displayed, because it contains errors.

Click for details.

The “diamond” version of the
24-cell seems unrelated to the
second version that shows all
vertices and edges, yet the
second version is implicit,
or hidden, in the first.
Hence “possibility.”

Neither version of the 24-cell
seems related in any obvious
way to the 3×3 grid, yet both
versions are implicit,
or hidden, in the grid.
Hence “possibility.”

Saturday, December 2, 2006

Saturday December 2, 2006

Filed under: General,Geometry — m759 @ 1:29 am

Venus at
St. Anne's,
continued

In honor of
the film "Bobby,"
now playing.

("Venus at St. Anne's"
is the title of the final
chapter of
the C. S. Lewis classic
That Hideous Strength.)

Star and Diamond

Symbol of Venus
and
Symbol of Plato

Related symbols:

Marilyn Monroe

Representation of Plato's Academy

Click on pictures
for details related tp
the Feast of St. Anne
(July 26).

"The best theology today,
in its repudiation of a
rhetorical religious idealism,
finds itself in agreement
with a recurrent note
in contemporary poetry….

We keep coming back
and coming back/
To the real: to the hotel
instead of the hymns/
That fall upon it
out of the wind.  We seek/
… Nothing beyond reality.
Within it/
Everything,
the spirit’s alchemicana….

(From 'An Ordinary Evening
in New Haven,'
in The Collected Poems
of Wallace Stevens….)

… Not grim/
Reality, but reality grimly seen….

(Ibid.)"

— "The Church's
New Concern with the Arts,"
by Amos N. Wilder,
Hollis Professor
of Divinity, Emeritus,
at Harvard Divinity School,
in Christianity and Crisis,
February 18, 1957.

 

 

 

"All the truth in the world
adds up to one big lie."

— Dylan, "Things Have Changed"
 

Wednesday, November 22, 2006

Wednesday November 22, 2006

Filed under: General,Geometry — m759 @ 12:00 pm

Windmill and Diamond

From “Today in History,”
by The Associated Press:

On this date:

In 1965, the musical
“Man of La Mancha”
opened in New York.

In 1975, Juan Carlos
was proclaimed
King of Spain.

Today’s birthdays:

… Movie director
Arthur Hiller is 83….

Hiller directed the 1972 film
of “Man of La Mancha.”

A quotation from that film:

“When life itself seems lunatic,
who knows where madness lies?”

Adapted from Log24 entries of
Jan. 5, 2003, and Feb. 1, 2003:

The image “http://www.log24.com/log/pix06B/061122-TimeEternity.gif” cannot be displayed, because it contains errors.

One can approach these symbols in either a mathematical or a literary fashion. For a mathematical discussion of the symbols’ structure, see Theme and Variations. Those who prefer literary discussions may make up their own stories.

 
“Plato is wary of all forms of rapture other than reason’s. He is most deeply leery of, because himself so susceptible to, the literary imagination. He speaks of it as a kind of holy madness or intoxication and goes on to link it to Eros, another derangement that joins us, but very dangerously, with the gods.”
 
— Rebecca Goldstein in
    The New York Times,
    December 16, 2002 
 
“It’s all in Plato, all in Plato;
bless me, what do they
teach them at these schools?”
 
— C. S. Lewis in
The Narnia Chronicles

Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 pm
To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
 would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."


The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.


"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.
 

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
 experiences, ideas and feelings
into something
whole, committed
 and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract."

Wednesday, July 26, 2006

Wednesday July 26, 2006

Filed under: General,Geometry — m759 @ 7:20 pm

Venus at St. Anne’s
(Title of the closing chapter
of That Hideous Strength)

Star and Diamond

Symbol of Venus
and
Symbol of Plato

“What do they
teach them
   at these schools?”
— C. S. Lewis

Today is the
feast of St. Anne.

Tuesday, July 11, 2006

Tuesday July 11, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:11 pm

Not Crazy Enough?

Some children of the sixties may feel that today's previous two entries, on Syd Barrett, the Crazy Diamond, are not crazy enough.  Let them consult the times of those entries– 2:11 and 8:15– and interpret those times, crazily, as dates: 2/11 and 8/15.

This brings us to Stephen King territory– apparently the natural habitat of Syd Barrett.

See Log24 on a 2/11, Along Came a Dreamcatcher, and Log24 on an 8/15, The Line.

From 8/15, a remark of Plato:

"There appears to be a sort of war of Giants and Gods going on…"

(Compare with the remarks by Abraham Cowley for Tom Stoppard's recent birthday.)

From 2/11, two links: Halloween Meditations  and We Are the Key.

From Dreamcatcher (the film and the book):

The image “http://www.log24.com/log/pix06/060211-Freeman2.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06/060324-Dreamcatcher.gif” cannot be displayed, because it contains errors.

For Syd Barrett as Duddits,

The image “http://www.log24.com/log/pix06A/060711-Duddits.jpg” cannot be displayed, because it contains errors.

see Terry Kirby on Syd Barrett
(edited– as in Stephen King
and the New Testament—
for narrative effect):

"He appeared as the Floyd performed the song 'Shine On You Crazy Diamond.' It contains the words: 'Remember when you were young, you shone like the sun. Shine on you crazy diamond. Now there's a look in your eyes, like black holes in the sky.'

At first, they didn't recognise the man, whose head and eyebrows were shaved….

But this was the 'crazy diamond' himself: Syd Barrett, the subject of the song….

When Roger Waters saw his old friend, he broke down….

Rick Wright, the keyboards player, later told an interviewer:

… 'Roger [Waters] was in tears, I think I was; we were both in tears. It was very shocking… seven years of no contact and then to walk in while we're actually doing that particular track. I don't know – coincidence, karma, fate, who knows? But it was very, very, very powerful.'"

Remarks suitable for Duddits's opponent, Mister Gray, may be found in the 1994 Ph.D. thesis of Noel Gray.

"I refer here to Plato's utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave."

Plato's Diamond

Shine on, gentle Duddits.

Wednesday, May 17, 2006

Wednesday May 17, 2006

Filed under: General — Tags: — m759 @ 4:29 am

Tombstone

From today's New York Times:

Obituary—

"Jiri Frel, a mercurial and eccentric curator who helped build the J. Paul Getty Museum into a major center for Greek and Roman art but resigned after revelations about unscrupulous acquisition practices, died on April 29. He was 82…. a well-regarded expert in Greek tombstones…."

News story—

"ATHENS, May 16 — After four hours of talks here with the Greek culture minister, the director of the J. Paul Getty Museum in Los Angeles said Tuesday that he would press for the return of some of the Getty's most prized ancient artifacts to Greece…. Greece is seeking the repatriation of a… tombstone…."

From a photo accompanying the obituary:

The image “http://www.log24.com/log/pix06A/060517-Window.jpg” cannot be displayed, because it contains errors.
Museum
window

 

The image “http://www.log24.com/log/pix06A/060517-StarAndDiamond.bmp” cannot be displayed, because it contains errors.

To Aster, from Plato

Asteras eisathreis, Aster emos.
Eithe genoimen ouranos,
'os pollois ommasin eis se blepo.

You gaze at stars, my Star.
Would that I were born the starry sky,
that I with many eyes might gaze at you.

Related material:

Log24 entries of Dec. 31, 2002

Why Me?

Plato's Diamond

The Halmos Tombstone

Tuesday, April 25, 2006

Tuesday April 25, 2006

Filed under: General,Geometry — m759 @ 3:09 pm

“There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to
Richard J. Trudeau’s remarks on
the “Story Theory” of truth
as opposed to
the “Diamond Theory” of truth
in The Non-Euclidean Revolution

A Serious Position

“‘Teitelbaum,’ in German,
is ‘date palm.'”
— Generations, Jan. 2003   

“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”

— Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church)

From Nextbook.org, “a gateway to Jewish literature, culture, and ideas”:

NEW BOOKS: 02.16.05
Proofs and Paradoxes
Alfred Teitelbaum changed his name to Tarski in the early 20s, the same time he changed religions, but when the Germans invaded his native Poland, the mathematician was in California, where he remained. His “great achievement was his audacious assault on the notion of truth,” says Martin Davis, focusing on the semantics and syntax of scientific language. Alfred Tarski: Life and Logic, co-written by a former student, Solomon Feferman, offers “remarkably intimate information,” such as abusive teaching and “extensive amorous involvements.”

From Wikipedia, an unsigned story:

“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”

The image “http://www.log24.com/log/pix06/060425-Tarski.jpg” cannot be displayed, because it contains errors.

Alfred Tarski

The image “http://www.log24.com/log/pix06/060424-Crimson2.jpg” cannot be displayed, because it contains errors.

See also
 
The Crimson Passion.

Monday, March 13, 2006

Monday March 13, 2006

Filed under: General — Tags: , , — m759 @ 1:01 pm
Christ at the
Lapin Agile

The image “http://www.log24.com/log/pix06/060313-WasleyChrist1.jpg” cannot be displayed, because it contains errors.

“The Christ by Wasley,
on wall under his right arm
the Picasso painting.”

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

Related material from Log24:

Religious Symbolism at Princeton,
on the Nassau Church,

Counting Crows on
 the Feast of St. Luke
(“Fullness… Multitude”),

The Quality of Diamond,
in memory of
Saint Hans-Georg Gadamer,
who died at 102
four years ago on this date,

and

Diamonds Are Forever.

Friday, December 16, 2005

Friday December 16, 2005

Filed under: General,Geometry — m759 @ 2:00 pm
Jesus vs. the Goddess:
A Brief Chronology

In 1946, Robert Graves published “King Jesus, an historical novel based on the theory and Graves’ own historical conjecture that Jesus was, in fact, the rightful heir to the Israelite throne… written while he was researching and developing his ideas for The White Goddess.”

In 1948, C. S. Lewis finished the first draft of The Lion, The Witch, and The Wardrobe, a novel in which one of the main characters is “the White Witch.”

In 1948, Robert Graves published The White Goddess.

In 1949, Robert Graves published Seven Days in New Crete [also titled Watch the North Wind Rise], “a novel about a social distopia in which Goddess worship is (once again?) the dominant religion.”

Lewis died on November 22, 1963, the day John F. Kennedy was killed.

Related material:
Log24, December 10, 2005

Graves died on December 7 (Pearl Harbor Day), 1985.

Related material:
Log24, December 7, 2005, and
Log24, December 11, 2005

Jesus died, some say, on April 7 in the year 30 A.D.

Related material:

Art Wars, April 7, 2003:
Geometry and Conceptual Art,

Eight is a Gate, and

The image “http://www.log24.com/log/pix05B/051216-PlatoDiamond.gif” cannot be displayed, because it contains errors.

Plato’s Diamond

The image “http://www.log24.com/log/pix05B/051216-Motto.jpg” cannot be displayed, because it contains errors.

— Motto of
Plato’s Academy

“Plato is wary of all forms of rapture other than reason’s. He is most deeply leery of, because himself so susceptible to, the literary imagination. He speaks of it as a kind of holy madness or intoxication and goes on to link it to Eros, another derangement that joins us, but very dangerously, with the gods.”
 
— Rebecca Goldstein in
    The New York Times,
    three years ago today
    (December 16, 2002) 
 
“It’s all in Plato, all in Plato;
 bless me, what do they
teach them at these schools?”
 
— C. S. Lewis in
the Narnia Chronicles

“How much story do you want?”
— George Balanchine

Thursday, May 26, 2005

Thursday May 26, 2005

Filed under: General,Geometry — Tags: — m759 @ 4:23 pm

Drama of the Diagonal
"The beautiful in mathematics
resides in contradiction.
Incommensurability, logoi alogoi, was
the first splendor in mathematics."
— Simone Weil, Oeuvres Choisies,
éd. Quarto , Gallimard, 1999, p. 100
 

Logos Alogos
by S. H. Cullinane

"To a mathematician, mathematical entities have their own existence, they habitate spaces created by their intention.  They do things, things happen to them, they relate to one another.  We can imagine on their behalf all sorts of stories, providing they don't contradict what we know of them.  The drama of the diagonal, of the square…"

— Dennis Guedj, abstract of "The Drama of Mathematics," a talk to be given this July at the Mykonos conference on mathematics and narrative.

For the drama of the diagonal of the square, see

Thursday May 26, 2005

Filed under: General,Geometry — m759 @ 4:00 am
The Changing

The previous entry dealt with a transformation
of the diamond figure from Plato’s Meno
into a visual proof of the Pythagorean theorem:

The image “http://www.log24.com/theory/images/DiamondTurning.jpg” cannot be displayed, because it contains errors.

Here is a transformation of Plato’s diamond
into the “gyronny” pattern of heraldry:

The image “http://www.log24.com/theory/images/Gyronny.gif” cannot be displayed, because it contains errors.
Viking Heraldry

The image “http://www.log24.com/theory/images/DiamondChanging2.jpg” cannot be displayed, because it contains errors.

For the mathematics dealing with
this sort of transformation, see
The Diamond 16 Puzzle and Diamond Theory.

Wednesday, May 25, 2005

Wednesday May 25, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:22 pm

The Turning

Readers who have an Amazon.com account may view book pages relevant to the previous entry.  See page 77 of The Way We Think, by Fauconnier and Turner (Amazon search term = Meno).  This page discusses both the Pythagorean theorem and Plato's diamond figure in the Meno, but fails to "blend" these two topics.  See also page 53 of The History of Mathematics, by Roger Cooke (first edition), where these two topics are in fact blended (Amazon search term = Pythagorean).  The illustration below is drawn from the Cooke book.

The image “http://www.log24.com/log/pix05/050525-Figs.jpg” cannot be displayed, because it contains errors.

Cooke demonstrates how the Pythagorean theorem might have been derived by "blending" Plato's diamond (left) with the idea of moving the diamond's corners (right).

The previous entry dealt with a conference on mathematics and narrative.  Above is an example I like of mathematics…. Here is an example I like of narrative: 

Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn  certain  that
it  was  the  sort of experience that her mother would not have
approved of on a first date.
     "Is this all part of what we have to do to go to  Asgard?"
she said. "Or are you just fooling around?"
     "We will go to Asgard...now," he said.
     At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect  was as if he had twisted the entire world through a
billionth part of a billionth  part  of  a  degree.  Everything
shifted,  was  for  a  moment  minutely  out of focus, and then
snapped back again as a suddenly different world.

— Douglas Adams, The Long Dark Tea-Time of the Soul

And here is a blend of the concepts "Asgard" and "conference":

"Asgard
    During the Interuniverse Society conference,
    a bridge was opened to Valhalla…."

  Bifrost
     In Norse myth, the rainbow bridge
     that connected Earth to Asgard,
     home of the gods.  It was extended
     to Tellus Tertius during the
     Interuniverse Society conference"

— From A Heinlein Concordance

The image “http://www.log24.com/log/pix05/050525-Rainbow.jpg” cannot be displayed, because it contains errors.

 

— Front page picture from a
local morning newspaper published
today, Wednesday, May 25, 2005

 

As George Balanchine once asked,
"How much story do you want?"

Friday, May 20, 2005

Friday May 20, 2005

Filed under: General — Tags: , — m759 @ 1:20 pm

The Shining of Apollo

"Plato's most significant passage may be found in Phaedrus 265b: 'And we made four divisions of the divine madness, ascribing them to four gods, saying that prophecy was inspired by Apollo, the mystic madness by Dionysos, the poetic by the Muses, and the madness of love […] by Aphrodite and Eros' (trans. by H.N. Fowler, in the Loeb Classical Library)."

— Saverio Marchignoli, note on section 20, paragraphs 115-119, of the Discourse on the Dignity of Man (Oratio de hominis dignitate) (1486) by Pico della Mirandola, considered the "Manifesto of the Renaissance."

Related material:
A Mass for Lucero,
The Shining of May 29,
Shining Forth,
Sermon for St. Patrick's Day, and the phrase
Diamond Struck by the Sun.

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:01 pm

Make a Différance

From Frida Saal's
Lacan The image “http://www.log24.com/log/pix05/050322-Diamond.gif” cannot be displayed, because it contains errors. Derrida:

"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names. For that reason their respective works become totally non-superposable and at the same time they were built with an awareness, or at least a partial awareness, of each other. What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."

 


From a Contemporary Literary Theory website:

"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."

From a text purchased on
Make a Difference Day, Oct. 23, 1999:

The image “http://www.log24.com/log/pix05/050322-Fig39.gif” cannot be displayed, because it contains errors.22. Without using the Pythagorean Theorem prove that the hypotenuse of  an isosceles right triangle will have the length The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.  if the equal legs have the length 1.  Suggestion: Consider the similar triangles in Fig. 39.
23.  The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas.  We cannot do so now because we have not yet considered the idea of area.  Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors. .

 

— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941)



Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics*  — they may in other respects be quite superior to men:

The image “http://www.log24.com/log/pix05/050322-Reba2.jpg” cannot be displayed, because it contains errors.

The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.

* See the remarks of Frida Saal above and of Barbara Johnson on mathematics (The Shining of May 29, cited in Readings for St. Patrick's Day).


† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005.  See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: , , , — m759 @ 2:20 pm

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 (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

— Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"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)

Thursday, January 27, 2005

Thursday January 27, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:29 am
Crystal Night

From artbook.com:

Mies van der Rohe:
Mies in Berlin

The image “http://www.log24.com/log/pix05/050127-Mies.jpg” cannot be displayed, because it contains errors.

Winner of
The Society of Architectural Historians
2002 Philip Johnson Award
for Excellence

Exhibition Catalog

"Published to accompany
a groundbreaking 2001 exhibition at
The Museum of Modern Art, New York."

 

From Mies and the Mastodon,
by Martin Filler, The New Republic,
issue dated Aug. 6, 2001:

"It would have been wiser for the new MoMA catalog… to have addressed the issue of his politics…. By ignoring such a central subject… the show gives off a mild stench of cover-up…. Only the German-born Rosemarie Haag Bletter (full disclosure: she is my wife) alludes to the verboten topic in her [catalog] essay on Mies's flirtation with crystal imagery, drawing a sharp parallel between the architect's extensive use of Kristallglas (plate glass) and the ensuing devastation of Kristallnacht, which erupted just three months after he left for the States."

Also from Filler's essay:

"Mies's rigorously simplified structures, typified by grids of steel and glass and an absence of applied ornament, represented the Platonic ideal of modernism for many people."

For more on history, politics, and
Mies's disciple Philip Johnson,
who died Tuesday evening, see

"We Cannot Not Know History."

For more on aesthetics, see the
Log24.net entry of Tuesday noon,

Diamonds Are Forever.

For more on a Platonic ideal of sorts,
see the following figure in two versions:
 
Version A, from Plato's Meno and
Diamond Theory,

The image “http://www.log24.com/log/pix05/050127-MenoDiamond.gif” cannot be displayed, because it contains errors.

and Version B,

The image “http://www.log24.com/log/pix05/050125-Forever.gif” cannot be displayed, because it contains errors.

from the date of Johnson's death
at his "famous crystalline box."

Was less more?

Friday, September 17, 2004

Friday September 17, 2004

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 pm

God is in…
The Details

From an entry for Aug. 19, 2003 on
conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

On Harvard and psychiatry: see

The Crimson Passion:
A Drama at Mardi Gras
(February 24, 2004)

This is a reductio ad absurdum of the Harvard philosophy so eloquently described by Alston Chase in his study of Harvard and the making of the Unabomber, Ted Kaczynski.  Kaczynski's time at Harvard overlapped slightly with mine, so I may have seen him in Cambridge at some point.  Chase writes that at Harvard, the Unabomber "absorbed the message of positivism, which demanded value-neutral reasoning and preached that (as Kaczynski would later express it in his journal) 'there is no logical justification for morality.'" I was less impressed by Harvard positivism, although I did benefit from a course in symbolic logic from Quine.  At that time– the early 60's– little remained at Harvard of what Robert Stone has called "our secret culture," that of the founding Puritans– exemplified by Cotton and Increase Mather.

From Robert Stone, A Flag for Sunrise:

"Our secret culture is as frivolous as a willow on a tombstone.  It's a wonderful thing– or it was.  It was strong and dreadful, it was majestic and ruthless.  It was a stranger to pity.  And it's not for sale, ladies and gentlemen."

Some traces of that culture:

A web page
in Australia:

A contemporary
Boston author:

Click on pictures for details.

A more appealing view of faith was offered by PBS on Wednesday night, the beginning of this year's High Holy Days:

Armand Nicholi: But how can you believe something that you don't think is true, I mean, certainly, an intelligent person can't embrace something that they don't think is true — that there's something about us that would object to that.

 

Jeremy Fraiberg: Well, the answer is, they probably do believe it's true.

Armand Nicholi: But how do they get there? See, that's why both Freud and Lewis was very interested in that one basic question. Is there an intelligence beyond the universe? And how do we answer that question? And how do we arrive at the answer of that question?

Michael Shermer: Well, in a way this is an empirical question, right? Either there is or there isn't.

Armand Nicholi: Exactly.

Michael Shermer: And either we can figure it out or we can't, and therefore, you just take the leap of faith or you don't.

Armand Nicholi: Yeah, now how can we figure it out?

Winifred Gallagher: I think something that was perhaps not as common in their day as is common now — this idea that we're acting as if belief and unbelief were two really radically black and white different things, and I think for most people, there's a very — it's a very fuzzy line, so that —

Margaret Klenck: It's always a struggle.

Winifred Gallagher: Rather than — I think there's some days I believe, and some days I don't believe so much, or maybe some days I don't believe at all.

Doug Holladay: Some hours.

Winifred Gallagher: It's a, it's a process. And I think for me the big developmental step in my spiritual life was that — in some way that I can't understand or explain that God is right here right now all the time, everywhere.

Armand Nicholi: How do you experience that?

Winifred Gallagher: I experience it through a glass darkly, I experience it in little bursts. I think my understanding of it is that it's, it's always true, and sometimes I can see it and sometimes I can't. Or sometimes I remember that it's true, and then everything is in Technicolor. And then most of the time it's not, and I have to go on faith until the next time I can perhaps see it again. I think of a divine reality, an ultimate reality, uh, would be my definition of God.

Winifred
Gallagher

Sangaku

Gallagher seemed to be the only participant in the PBS discussion that came close to the Montessori ideals of conciseness, simplicity, and objectivity.  Dr. Montessori intended these as ideals for teachers, but they seem also to be excellent religious values.  Just as the willow-tombstone seems suited to Geoffrey Hill's style, the Pythagorean sangaku pictured above seems appropriate to the admirable Gallagher.

Sunday, September 5, 2004

Sunday September 5, 2004

Filed under: General — m759 @ 4:00 pm

Symmetry and Change
in the Dreamtime

Notes from the Journal
of Steven H. Cullinane

Summary:

Aug 31 2004 
07:31:01 PM		 Early Evening,
Shining Star 
Sep 01 2004
09:00:35 AM 		Words
and Images
Sep 01 2004
12:07:28 PM 		Whale Rider
Sep 02 2004
11:11:42 AM 		Heaven
and Earth
Sep 02 2004
07:00:23 PM 		Whale Road

Sep 03 2004
12:00:54 AM

Cinderella’s
Slipper 		 
Sep 03 2004
10:01:56 AM 		Another
September Morn
  

Sep 03 2004
12:00:25 PM

Noon

Sep 03 2004
01:13:49 PM

De Nada

Sep 03 2004
03:17:13 PM

Ite, Missa Est 

Tuesday, August 31, 2004

Symmetry and Change, Part 1…

Early Evening,
Shining Star

7:31:01 PM ET

Hexagram 01
The Creative:

 

The Image

Heaven
Heaven

The movement of heaven
is full of power.

Click on picture
for details.

The Clare Lawler Prize
for Literature goes to…

Under the Volcano,
Chapter VI:

“What have I got out of my life? Contacts with famous men… The occasion Einstein asked me the time, for instance. That summer evening…. smiles when I say I don’t know. And yet asked me. Yes: the great Jew, who has upset the whole world’s notions of time and space, once leaned down… to ask me… ragged freshman… at the first approach of the evening star, the time. And smiled again when I pointed out the clock neither of us had noticed.”

For the thoughts on time
of another famous man,
from Mexico, see the
Nobel Prize acceptance speech
of Octavio Paz,
“In Search of the Present.”

Wednesday, September 1, 2004

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire
Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

“As the quotes above by Nabokov and Welles suggest, we need to be able to account for the specific functions available to narrative in each medium, for the specific elements that empirical creators will ‘play with’ in crafting their narratives.”

— Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

Stay for just a while…
Stay, and let me look at you.
It’s been so long, I hardly knew you.
Standing in the door…
Stay with me a while.
I only want to talk to you.
We’ve traveled halfway ’round the world
To find ourselves again.

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

“In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

I personally enjoyed this site’s description of Billy Crystal’s remarks, which included “a joke about former President Clinton’s forthcoming children’s book — ‘It’s called The Little Engine That Could Because It Could.'”

“Puff, puff, woo, woo, off we go!” 

 

 

Wednesday, September 1, 2004

Symmetry and Change, Part 3…

Whale Rider

12:07:28 PM

Hexagram 28
Preponderance of
the Great:

The Image

Lake
Wind

The lake rises
above the trees.

 

Cullinane College News:

“Congratulations to Clare Lawler, who participated very successfully in the recently held Secondary Schools Judo Championships in Wellington.”

For an explanation of this entry’s title, see the previous two entries and

Oxford Word
(Log24, July 10, 2004) 

Thursday, September 2, 2004

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind
Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”


Kylie


Finney

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.
— Lonnie Donegan (d. Nov. 3)
and others
The Rock Island Line’s namesake depot 
in Rock Island, Illinois

“What it all boiled down to really was everybody giving everybody else a hard time for no good reason whatever… You just couldn’t march to your own music. Nowadays, you couldn’t even hear it… It was lost, the music which each person had inside himself, and which put him in step with things as they should be.”

— The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing”

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star. 

Thursday, September 2, 2004

Symmetry and Change, Part 5…

Whale Road

7:00:23 PM

Hexagram 23
Splitting Apart:

The Image

Mountain
Earth

The mountain rests
on the earth.

“… the plot is different but the monsters, names, and manner of speaking will ring a bell.”

— Frank Pinto, Jr., review of Seamus Heaney’s translation of Beowulf 

Other recommended reading, found during a search for the implications of today’s previous entry, “Hexagram 42”:

Water Wings.

This excellent meditation
on symmetry and change
comes from a site whose
home page
has the following image:

Friday, September 3, 2004

 Symmetry and Change, Part 6…

Cinderella’s Slipper

12:00:54 AM ET

Hexagram 54
The Marrying Maiden:

 

The Image

Thunder

Lake
See
The hundredletter
thunderwords of
Finnegans Wake

“… a Thoreau-like retreat
by a nearby lake….
Both men have
a ‘touch of the poet’….
The symmetry is perfect.”

Friday, September 3, 2004  

Symmetry and Change, Part 7…

Another September Morn

10:01:56 AM ET

Hexagram 56:
The Wanderer

 

The Image

Fire

Mountain

Fire on the mountain,
Run boys run…
Devil’s in the House of
The Rising Sun! 

Friday, September 3, 2004

Symmetry and Change, Part 8…

Noon

12:00:25 PM ET

Hexagram 25
Innocence:

The Image

Heaven

Thunder

Under heaven
thunder rolls. 

Friday, September 3, 2004

Symmetry and Change, Part 9…

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

Lake

Fire
 Fire in the lake:
the image of Revolution.

“I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.”

— Malcolm Lowry, Under the Volcano 

Friday, September 3, 2004

Symmetry and Change, conclusion…

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven

Fire

Heaven together with fire.

“A pretty girl —
is like a melody —- !”

 For details, see
A Mass for Lucero. 

Thursday, September 2, 2004

Thursday September 2, 2004

Filed under: General — m759 @ 11:11 am

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind
Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”


Kylie


Finney

 
Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.
— Lonnie Donegan (d. Nov. 3)
and others
 
 
The Rock Island Line’s namesake depot 
in Rock Island, Illinois

“What it all boiled down to really was everybody giving everybody else a hard time for no good reason whatever… You just couldn’t march to your own music. Nowadays, you couldn’t even hear it… It was lost, the music which each person had inside himself, and which put him in step with things as they should be.”

— The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing“

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star.

Sunday, August 15, 2004

Sunday August 15, 2004

Filed under: General — m759 @ 3:17 pm

The Line

Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance, Ch. 6 (italics are mine):

“A classical understanding sees the world primarily as underlying form itself. A romantic understanding sees it primarily in terms of immediate appearance.”

The Sophist, by Plato:

STRANGER – We are far from having exhausted the more exact thinkers who treat of being and not-being. But let us be content to leave them, and proceed to view those who speak less precisely; and we shall find as the result of all, that the nature of being is quite as difficult to comprehend as that of not-being.

THEAETETUS – Then now we will go to the others.

STRANGER – There appears to be a sort of war of Giants and Gods going on amongst them; they are fighting with one another about the nature of essence.

THEAETETUS – How is that?

STRANGER – Some of them are dragging down all things from heaven and from the unseen to earth, and they literally grasp in their hands rocks and oaks; of these they lay hold, and obstinately maintain, that the things only which can be touched or handled have being or essence, because they define being and body as one, and if any one else says that what is not a body exists they altogether despise him, and will hear of nothing but body.

THEAETETUS – I have often met with such men, and terrible fellows they are.

STRANGER – And that is the reason why their opponents cautiously defend themselves from above, out of an unseen world, mightily contending that true essence consists of certain intelligible and incorporeal ideas; the bodies of the materialists, which by them are maintained to be the very truth, they break up into little bits by their arguments, and affirm them to be, not essence, but generation and motion. Between the two armies, Theaetetus, there is always an endless conflict raging concerning these matters.

THEAETETUS – True.

— Translated by Benjamin Jowett

Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance, Ch. 18:

“The wave of crystallization rolled ahead. He was seeing two worlds, simultaneously. On the intellectual side, the square side, he saw now that Quality was a cleavage term. What every intellectual analyst looks for. You take your analytic knife, put the point directly on the term Quality and just tap, not hard, gently, and the whole world splits, cleaves, right in two…

The Line,
by S. H. Cullinane

hip and square, classic and romantic, technological and humanistic…and the split is clean. There’s no mess. No slop. No little items that could be one way or the other. Not just a skilled break but a very lucky break. Sometimes the best analysts, working with the most obvious lines of cleavage, can tap and get nothing but a pile of trash. And yet here was Quality; a tiny, almost unnoticeable fault line; a line of illogic in our concept of the universe; and you tapped it, and the whole universe came apart, so neatly it was almost unbelievable. He wished Kant were alive. Kant would have appreciated it. That master diamond cutter. He would see. Hold Quality undefined. That was the secret.”

What Pirsig means by “quality” is close to what Yagoda means, in the previous entry, by “style.”

Monday, April 5, 2004

Monday April 5, 2004

Filed under: General,Geometry — Tags: , , — m759 @ 4:03 am

Ideas and Art

 
— Motto of
Plato's Academy

 

From Minimalist Fantasies,
by Roger Kimball, May 2003:

All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
—Frank Stella, 1966

Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
— Clement Greenberg, 1967

The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
— Michael Kimmelman, 2003

There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics.
— Roger Kimball, 2003

 

 

From How Not Much Is a Whole World,
by Michael Kimmelman, April 2, 2004

Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it


"William Clark,"
by Patricia Johanson, 1967

should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it.

….

The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line.

If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence….

That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one.

 

I maintain that of course
we can see ideas.

Example: the idea of
invariant structure.

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235.
    Pergamon Press Ltd., 1978

For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.

Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)…  That is to say, they are not lost in translation.  (See previous entry.)

Sunday, March 7, 2004

Sunday March 7, 2004

Filed under: General — Tags: , — m759 @ 6:00 pm

Apartments

From Wallace Stevens,
"Notes Toward a Supreme Fiction":

It is the celestial ennui of apartments
That sends us back to the first idea, the quick
Of this invention; and yet so poisonous

Are the ravishments of truth, so fatal to
The truth itself, the first idea becomes
The hermit in a poet’s metaphors,

Who comes and goes and comes and goes all day.
May there be an ennui of the first idea?
What else, prodigious scholar, should there be?….

From Guyan Robertson,
Groups Acting on Affine Buildings
and their Boundaries:

From Plato's Meno:

They will get it straight one day at the Sorbonne.
We shall return at twilight from the lecture         
Pleased that the irrational is rational….              

See Logos and Logic
and the previous entry.

Sunday, February 8, 2004

Sunday February 8, 2004

Filed under: General,Geometry — m759 @ 2:00 pm

The Quality of Diamond

On February 3, 2004, archivist and abstract painter Ward Jackson died at 75.  From today’s New York Times:

“Inspired by painters like Piet Mondrian and Josef Albers, Mr. Jackson made austere, hard-edged geometric compositions, typically on diamond-shaped canvases.”

On a 2003 exhibit by Pablo Helguera that included Mr. Jackson:

Parallel Lives

Parallel Lives recounts and recontextualizes real episodes from the lives of five disparate individuals including Florence Foster Jenkins, arguably the world’s worst opera singer; Giulio Camillo, a Renaissance mystic who aimed to build a memory container for all things; Friedrich Froebel, the inventor of the kindergarten education system, the members of the last existing Shaker community, and Ward Jackson, the lifelong archivist of the Solomon R. Guggenheim Museum.

Parallel Lives pays homage to Hans-Georg Gadamer (1900-2002) and his system of philosophical hermeneutics built through an exploration of historicity, language, and art. This exhibition, which draws its title from the classic work by Plutarch, is a project that explores biography as a medium, drawing from the earlier innovation of the biographical practice in works like Marcel Schwob’s “Imaginary Lives” (1896) and John Aubrey’s “Brief Lives” (1681). Through display means, the project blends the lives of these individuals into one basic story, visually stating the relationship between individualism and society as best summarized by Gadamer’s famous phrase: “we all are others, and we all are a self.”

On February 3, the day that Jackson died, there were five different log24.net entries:

  1. The Quality with No Name 
  2. Speaking Globally
  3. Lila
  4. Theory of Design
  5. Retiring Faculty.

Parallels with the Helguera exhibit:

Florence Foster Jenkins: Janet Jackson in (2) above.

Giulio Camillo: Myself as compiler of the synchronistic excerpts in (5).

Friedrich Froebel: David Wade in (4).

The last Shakers: Christopher Alexander and his acolytes in (1).

Ward Jackson: On Feb. 3, Jackson became a permanent part of Quality — i.e., Reality — itself, as described in (3).

Some thoughts of Hans-Georg Gadamer
relevant to Jackson’s death:

Gadamer, Art, and Play

by G.T. Karnezis

The pleasure it [art] elicits “is the joy of knowledge.” It does not operate as an enchantment but “a transformation into the true.” Art, then, would seem to be an essentializing agent insofar as it reveals what is essential. Gadamer asks us to see reality as a horizon of “still undecided possibilities,” of unfulfilled expectations, of contingency. If, in a particular case, however, “a meaningful whole completes and fulfills itself in reality,” it is like a drama. If someone sees the whole of reality as a closed circle of meaning” he will be able to speak “of the comedy and tragedy of life” (genres becoming ways of conceiving reality). In such cases where reality “is understood as a play, there emerges the reality of what play is, which we call the play of art.” As such, art is a realization: “By means of it everyone recognizes that that is how things are.” Reality, in this viewpoint, is what has not been transformed. Art is defined as “the raising up of this reality to its truth.”

As noted in entry (3) above
on the day that Jackson died,

“All the world’s a stage.”

— William Shakespeare

Tuesday, August 19, 2003

Tuesday August 19, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:23 pm

Intelligence Test

From my August 31, 2002, entry quoting Dr. Maria Montessori on conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life. 

 

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

For the meaning of life, see

The Gospel According to Jill St. John,

whose birthday is today.

"Some sources credit her with an I.Q. of 162."
 

Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: , — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Further Reading:

Friday, July 25, 2003

Friday July 25, 2003

Filed under: General — Tags: , , — m759 @ 5:24 pm

For Jung’s 7/26 Birthday:
A Logocentric Meditation

Leftist academics are trying to pull a fast one again.  An essay in the most prominent American mathematical publication tries to disguise a leftist attack on Christian theology as harmless philosophical woolgathering.

In a review of Vladimir Tasic’s Mathematics and the Roots of Postmodern Thought, the reviewer, Michael Harris, is being less than candid when he discusses Derrida’s use of “logocentrism”:

“Derrida uses the term ‘logocentrism’… as ‘the metaphysics of phonetic writing’….”

— Notices of the American Mathematical Society, August 2003, page 792

We find a rather different version of logocentrism in Tasic’s own Sept. 24, 2001, lecture “Poststructuralism and Deconstruction: A Mathematical History,” which is “an abridged version of some arguments” in Tasic’s book on mathematics and postmodernism:

“Derrida apparently also employs certain ideas of formalist mathematics in his critique of idealist metaphysics: for example, he is on record saying that ‘the effective progress of mathematical notation goes along with the deconstruction of metaphysics.’

Derrida’s position is rather subtle. I think it can be interpreted as a valiant sublation of two completely opposed schools in mathematical philosophy. For this reason it is not possible to reduce it to a readily available philosophy of mathematics. One could perhaps say that Derrida continues and critically reworks Heidegger’s attempt to ‘deconstruct’ traditional metaphysics, and that his method is more ‘mathematical’ than Heidegger’s because he has at his disposal the entire pseudo-mathematical tradition of structuralist thought. He has himself implied in an interview given to Julia Kristeva that mathematics could be used to challenge ‘logocentric theology,’ and hence it does not seem unreasonable to try looking for the mathematical roots of his philosophy.”

The unsuspecting reader would not know from Harris’s review that Derrida’s main concern is not mathematics, but theology.  His ‘deconstruction of metaphysics’ is actually an attack on Christian theology.

From “Derrida and Deconstruction,” by David Arneson, a University of Manitoba professor and writer on literary theory:

“Logocentrism: ‘In the beginning was the word.’ Logocentrism is the belief that knowledge is rooted in a primeval language (now lost) given by God to humans. God (or some other transcendental signifier: the Idea, the Great Spirit, the Self, etc.) acts a foundation for all our thought, language and action. He is the truth whose manifestation is the world.”

Some further background, putting my July 23 entry on Lévi-Strauss and structuralism in the proper context:

Part I.  The Roots of Structuralism

“Literary science had to have a firm theoretical basis…”

Part II.  Structuralism/Poststructuralism

“Most [structuralists] insist, as Levi-Strauss does, that structures are universal, therefore timeless.”

Part III.  Structuralism and
Jung’s Archetypes

Jung’s “theories, like those of Cassirer and Lévi-Strauss, command for myth a central cultural position, unassailable by reductive intellectual methods or procedures.”

And so we are back to logocentrism, with the Logos — God in the form of story, myth, or archetype — in the “central cultural position.”

What does all this have to do with mathematics?  See

Plato’s Diamond,

Rosalind Krauss on Art –

“the Klein group (much beloved of Structuralists)”

Another Michael Harris Essay, Note 47 –

“From Krauss’s article I learned that the Klein group is also called the Piaget group.”

and Jung on Quaternity:
Beyond the Fringe –

“…there is no denying the fact that [analytical] psychology, like an illegitimate child of the spirit, leads an esoteric, special existence beyond the fringe of what is generally acknowledged to be the academic world.”

What attitude should mathematicians have towards all this?

Towards postmodern French
atheist literary/art theorists –

Mathematicians should adopt the attitude toward “the demimonde of chic academic theorizing” expressed in Roger Kimball’s essay, Feeling Sorry for Rosalind Krauss.

Towards logocentric German
Christian literary/art theorists –

Mathematicians should, of course, adopt a posture of humble respect, tugging their forelocks and admitting their ignorance of Christian theology.  They should then, if sincere in their desire to honestly learn something about logocentric philosophy, begin by consulting the website

The Quest for the Fiction of an Absolute.

For a better known, if similarly disrespected, “illegitimate child of the spirit,” see my July 22 entry.

Saturday, July 5, 2003

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 7:21 pm

Elementary,
My Dear Gropius

“What is space, how can it be understood and given a form?”
— Walter Gropius

Stoicheia:

Stoicheia,” Elements, is the title of
Euclid’s treatise on geometry.

Stoicheia is apparently also related to a Greek verb meaning “march” or “walk.”

According to a website on St. Paul’s phrase “ta stoicheia tou kosmou,” which might be translated

The Elements of the Cosmos,

“… the verbal form of the root stoicheo was used to mean, ‘to be in a line,’ ‘to march in rank and file.’ … The general meaning of the noun form (stoicheion) was ‘what belongs to a series.’ “

As noted in my previous entry, St. Paul used a form of stoicheo to say “let us also walk (stoichomen) by the Spirit.” (Galatians 5:25) The lunatic ravings* of Saul of Tarsus aside, the concepts of walking, of a spirit, and of elements may be combined if we imagine the ghost of Gropius strolling with the ghosts of Plato, Aristotle, and Euclid, and posing his question about space.  Their reply might be along the following lines:

Combining stoicheia with a peripatetic peripateia (i.e., Aristotelian plot twist), we have the following diagram of Aristotle’s four stoicheia (elements),

which in turn is related, by the “Plato’s diamond” figure in the monograph Diamond Theory, to the Stoicheia, or Elements, of Euclid.

Quod erat demonstrandum.

* A phrase in memory of the Paulist Norman J. O’Connor, the “jazz priest” who died on St. Peter’s day, Sunday, June 29, 2003.  Paulists are not, of course, entirely mad; the classic The Other Side of Silence: A Guide to Christian Meditation, by the Episcopal priest Morton Kelsey, was published by the Paulist Press.

Its cover (above), a different version of the four-elements theme, emphasizes the important Jungian concept of quaternity.  Jung is perhaps the best guide to the bizarre world of Christian symbolism.  It is perhaps ironic, although just, that the Paulist Fathers should distribute a picture of “ta stoicheia tou kosmou,” the concept that St. Paul himself railed against.

The above book by Kelsey should not be confused with another The Other Side of Silence, a work on gay history, although confusion would be understandable in light of recent ecclesiastical revelations.

Let us pray that if there is a heaven, Father O’Connor encounters there his fellow music enthusiast Cole Porter rather than the obnoxious Saul of Tarsus.

Friday, April 25, 2003

Friday April 25, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 pm

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.

 


The rune known as
"Dagaz"

Represents
the balance point or "still point."


The Nike 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.
 

Thursday, March 27, 2003

Thursday March 27, 2003

Filed under: General — m759 @ 2:35 pm

Sixteen

war protesters, handcuffed together, blocked traffic near 47th Street and 5th Avenue in New York City Wednesday.  They chanted “Occupation is a Crime, Free Iraq and Palestine!”

— Newsday, March 26, 2003

Forty-seventh Street between Fifth and Sixth Avenues is known as the diamond district (or, in Buddhist parlance, “Diamond Way”).

“Is it safe?”
— Sir Laurence Olivier, 1976

Monday, March 24, 2003

Monday March 24, 2003

Filed under: General — m759 @ 12:52 pm

Orwell’s question, according to
an admirer of leftist Noam Chomsky:

“When so much of the BS is right out in the open,
why is it that we know so little about it?
Why don’t we see what’s right in front of our eyes?”


Oscar
Deep Chomsky:
Lying, Truth-Telling,
and the Social Order
 
 
 
 
 Michael
 Moore

“First of all, I’d like to thank the Academy….”
— Quotation attributed to Plato

The New Yorker of March 31, 2003, discusses leftist academic Noam Chomsky.  The online edition provides a web page listing pro-Chomsky links.

Chomsky’s influence is based in part on the popularity of his half-baked theories on linguistics, starting in the 1950’s with “deep structure” and “transformational,” or “generative,” grammar.

Chomsky has abandoned many of his previous ideas and currently touts what he calls The Minimalist Program.

For some background on Chomsky’s recent linguistic notions, see the expository essay “Syntactic Theory,” by Elly van Gelderen of the Arizona State University English Department.  Van Gelderen lists her leftist political agenda on her “Other Interests” page.  Her department may serve as an example of how leftists have converted many English departments in American universities to propaganda factories.

Some attacks on Chomsky’s scholarship:

The Emperor’s New Linguistics

The New Grammarians’ Funeral

Beyond Chomsky

Could Chomsky Be Wrong? 

Forty-four Reasons Why the Chomskians Are Mistaken

Call for Papers, Chomsky 2003

Chomsky’s (Mis)Understanding of Human Thinking

Anatomy of a Revolution… Chomsky in 1962

…Linguistic Theory: The Rationality of Noam Chomsky

A Bibliography

Some attacks on Chomsky’s propaganda:

LeftWatch.com Chomsky page

Destructive Generation excerpt

The Sick Mind of Noam Chomsky

Partners in Hate: Noam Chomsky and the Holocaust Deniers

Chomsky and Plato’s Diamond

Like another purveyor of leftist nonsense, Jacques Derrida, Chomsky is fond of citing Plato as a precedent.  In particular, what Chomsky calls “Plato’s problem” is discussed in Plato’s Meno.  For a look at the diamond figure that plays a central role in that dialogue, see Diamond Theory.  For an excellent overview of related material in Plato, see Theory of Forms.

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General — Tags: — m759 @ 4:44 pm

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.

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.

Wednesday, March 12, 2003

Wednesday March 12, 2003

Filed under: General,Geometry — m759 @ 2:03 am

Daimon Theory

Today is allegedly the anniversary of the canonization, in 1622, of two rather important members of the Society of Jesus (Jesuits):

Ignatius Loyola…
  Click here for Loyola’s legacy of strategic intelligence.

Francis Xavier…
  Click here for Xavier’s legacy of strategic stupidity.

We can thank (or blame) a Jesuit (Gerard Manley Hopkins) for the poetic phrase “immortal diamond.”  He may have been influenced by Plato, who has Socrates using a diamond figure in an argument for the immortality of the soul.  Confusingly, Socrates also talked about his “daimon” (pronounced dye-moan).  Combining these similar-sounding concepts, we have Doctor Stephen A. Diamond writing about daimons — a choice of author and topic that neatly combines the strategic intelligence of Loyola with the strategic stupidity of Xavier.

The cover illustration is perhaps not of Dr. Diamond himself.

A link between diamond theory and daimon theory is furnished by the charitable legacy of the non-practicing Jew Walter Annenberg.

For Annenberg and diamond theory, see this site on the elementary geometry of quilt blocks, which credits the Annenberg Foundation for support.

For Annenberg and daimon theory, see this site on Socrates, which has a similar Annenberg support credit.

Advanced disciples of Annenberg can learn much from the Perseus site about daimon theory. Let us pray that Abrahamic religious bigotry does not stand in their way.  Less advanced disciples of Annenberg may find fulfillment in teaching children the beauty of elementary 4×4 quilt-block symmetry.  Let us pray that academic bigotry does not prevent these same children, when they have grown older, from learning the deeper, and more difficult, beauties of diamond theory.

 
Daimon Theory

 
Diamond Theory

Sunday, January 5, 2003

Sunday January 5, 2003

Filed under: General,Geometry — Tags: — m759 @ 12:12 am

Whirligig

Thus the whirligig of time brings in his revenges.
Twelfth Night. Act v. Sc. 1.

Twelfth night is the night of January 5-6.

Tonight is twelfth night in Australia; 4 AM Jan. 5
in New York City is 8 PM Jan. 5 in Sydney.

An October 6 entry:

Twenty-first Century Fox

On Sunday, October 6, 1889, the Moulin Rouge music hall opened in Paris, an event that to some extent foreshadowed the opening of Fox Studios Australia in Sydney on November 7, 1999.  The Fox ceremonies included, notably, Kylie Minogue singing "Diamonds are a Girl's Best Friend." 

 

Red Windmill

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

An October 5 entry:

The Message from Vega

"Mercilessly tasteful"
 — Andrew Mueller,
review of Suzanne Vega's
"Songs in Red and Gray"

In accordance with the twelfth-night
"whirligig of time" theme,
here are two enigmatic quilt blocks:

Devil's Claws, or
Hourglass Var. 3

Yankee Puzzle, or
Hourglass Var. 5
 
One can approach these symbols in either a literary or a mathematical fashion. For a purely mathematical discussion of the differences in the two symbols' structure, see Diamond Theory. Those who prefer literary discussions may make up their own stories.
 
"Plato is wary of all forms of rapture other than reason's. He is most deeply leery of, because himself so susceptible to, the literary imagination. He speaks of it as a kind of holy madness or intoxication and goes on to link it to Eros, another derangement that joins us, but very dangerously, with the gods."
 
— Rebecca Goldstein in The New York Times,
    December 16, 2002 
 
"It's all in Plato, all in Plato; bless me,
what do they teach them at these schools?"
 
— C. S. Lewis in the Narnia Chronicles 

Monday, December 16, 2002

Monday December 16, 2002

Filed under: General,Geometry — m759 @ 10:00 pm

Rebecca Goldstein
at Heaven’s Gate

This entry is in gratitude for Rebecca Goldstein’s
excellent essay
in The New York Times of December 16, 2002.

She talks about the perennial conflict between two theories of truth that Richard Trudeau called the “story theory” and the “diamond theory.” My entry of December 13, 2002, “Rhyme Scheme,” links the word “real” to an article in the Stanford Encyclopedia of Philosophy that contains the following:

“According to a platonist about arithmetic, the truth of the sentence ‘7 is prime’ entails the existence of an abstract object, the number 7. This object is abstract because it has no spatial or temporal location, and is causally inert. A platonic realist about arithmetic will say that the number 7 exists and instantiates the property of being prime independently of anyone’s beliefs, linguistic practices, conceptual schemes, and so on. A certain kind of nominalist rejects the existence claim which the platonic realist makes: there are no abstract objects, so sentences such as ‘7 is prime’ are false…”

This discussion of “sevenness,” along with the discussion of “eightness” in my December 14, 2002, note on Bach, suggest that I supply a transcription of a note in my paper journal from 2001 that deals with these matters.

From a paper journal note of October 5, 2001:

The 2001 Silver Cup Award
for Realism in Mathematics
goes to…
Glynis Johns, star of
The Sword and the Rose,
Shake Hands with the Devil, and
No Highway in the Sky.

Glynis Johns is 78 today.

“Seven is heaven,
Eight is a gate.”
— from Dealing with Memory Changes
as You Grow Older,
by Kathleen Gose and Gloria Levi

“There is no highway in the sky.”
— Quotation attributed to Albert Einstein.
(See Gotthard Günther’s website
“Achilles and the Tortoise, Part 2”.)

“Don’t give up until you
Drink from the silver cup
And ride that highway in the sky.”
—America, 1974

See also page 78 of
Realism in Mathematics
(on Gödel’s Platonism)
by Penelope Maddy,
Clarendon Press, Oxford, 1990
(reprinted, 2000).

Added 12/17/02: See also
the portrait of Rebecca Goldstein in
Hadassah Magazine
 Volume
78
Number 10
(June/July 1997).

For more on the Jewish propensity to
assign mystical significance to numbers, see
Rabbi Zwerin’s Kol Nidre Sermon.

For the significance of “seven” in Judaism, see
Zayin: The Woman of Valor.
For the significance of “eight” in Judaism, see
Chet: The Life Dynamic.

For the cabalistic significance of
“Seven is heaven, Eight is a gate,”
note that Zayin, Seven, signifies
“seven chambers of Paradise”
and that Chet, Eight, signifies
the “gateway to infinity.”

For the significance of the date 12.17, see
Tet: The Concealed Good.

Saturday, November 9, 2002

Saturday November 9, 2002

Filed under: General,Geometry — m759 @ 4:44 am

Birthdate of Hermann Weyl

Weyl


Plato’s Diamond

Result of a Google search.

Category:  Science > Math > Algebra > Group Theory 

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

Sponsored Link

Symmetry Puzzle
New free online puzzle illustrates
the mathematics of symmetry.
m759.freeservers.com/puzzle.
html

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


“Shine On, 
   You Crazy Diamond.”   
 

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.

Thursday, October 31, 2002

Thursday October 31, 2002

Filed under: General,Geometry — m759 @ 11:07 pm

Plato's
Diamond

From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:

"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…" 

Charles H. Bennett has written about Omega as a cabalistic number.

Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).

From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:

"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."

From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:

SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….

From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:

Galileo on Plato's diamond:

"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"

Roger Bacon on Plato's diamond:

"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"

It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.

Sunday, September 15, 2002

Sunday September 15, 2002

Filed under: General,Geometry — Tags: — m759 @ 11:07 pm

Evariste Galois and 
The Rock That Changed Things

An article in the current New York Review of Books (dated Sept. 26) on Ursula K. Le Guin prompted me to search the Web this evening for information on a short story of hers I remembered liking.  I found the following in the journal of mathematician Peter Berman:

  • A Fisherman of the Inland Sea, Ursula K. Le Guin, 1994:
    A book of short stories. Good, entertaining. I especially liked “The Rock That Changed Things.” This story is set in a highly stratified society, one split between elite and enslaved populations. In this community, the most important art form is a type of mosaic made from rocks, whose patterns are read and interpreted by scholars from the elite group. The main character is a slave woman who discovers new patterns in the mosaics. The story is slightly over-the-top but elegant all the same.

I agree that the story is elegant (from a mathematician, a high compliment), so searched Berman’s pages further, finding this:

A table of parallels

between The French Mathematician (a novel about Galois) and Harry Potter and the Sorcerer’s Stone. 

My own version of the Philosopher’s Stone (the phrase used instead of “Sorcerer’s Stone” in the British editions of Harry Potter) appears in my profile picture at top left; see also the picture of Plato’s diamond figure in my main math website.  The mathematics of finite (or “Galois”) fields plays a role in the underlying theory of this figure’s hidden symmetries.  Since the perception of color plays a large role in the Le Guin story and since my version of Plato’s diamond is obtained by coloring Plato’s version, this particular “rock that changes things” might, I hope, inspire Berman to extend his table to include Le Guin’s tale as well.

Even the mosaic theme is appropriate, this being the holiest of the Mosaic holy days.

Dr. Berman, G’mar Chatimah Tova.

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