Log24

Friday, September 9, 2016

Notation

Filed under: General — Tags: , , — m759 @ 10:00 PM

The New York Times  interviews Alan Moore

“A version of this article appears in print on September 11, 2016,
on page BR9 of the Sunday Book Review ….”

“What genres do you prefer? And which do you avoid?”

“To be honest, having worked in genre for so long, I’m happiest
when I’m outside it altogether, or perhaps more accurately,
when I can conjure multiple genres all at once, in accordance
with my theory (now available, I believe, as a greeting card and
fridge magnet) that human life as we experience it is a
simultaneous multiplicity of genres. I put it much more elegantly
on the magnet.”

Saturday, August 27, 2016

Folk Notation

Filed under: General — Tags: , — m759 @ 2:01 PM

See the Chautauqua Season post of June 25
and a search for Notation  in this journal.

See as well the previous post and Bullshit Studies .

Saturday, August 14, 2010

Iconic Notation

Filed under: General — m759 @ 12:12 AM

Continued from Friday the 13th

(Click to enlarge.)

http://www.log24.com/log/pix10B/100814-DBsm.jpg

Related material—

The image “http://www.log24.com/log/pix07A/070814-timejoin15.jpg” cannot be displayed, because it contains errors.
Cover art by Barclay Shaw reprinted from an earlier (1984) edition

IMAGE- Variations on Hexagram 14

A question from Ivan Illich
(founder of CIDOC, the Center for Intercultural Documentation,
in Cuernavaca, Mexico)—

"Who can be served by bridges to nowhere?"

For more about nowhere, see Utopia. See also http://outis.blogspot.com.

Tuesday, August 17, 2021

Mythical Figure

Filed under: General — Tags: — m759 @ 12:55 PM

A phrase from the previous post : "modern society's mythical figures."

A mythical figure from Claude Lévi-Strauss

The above image is from a study of Lévi-Strauss's "Canonical Formula" …

Tuesday, July 21, 2020

Oeuvre

Filed under: General — m759 @ 5:13 AM
From “Nabokov’s Crosswords of Composition,” by
Rebecca Freeh-Maciorowski, a paper presented at NEMLA, dated 15 October 2014 —

“In a way, Nabokov’s entire oeuvre might be built upon one all-encompassing ‘crossword,’ a possibility raised by W.W. Rowe when he writes ‘Words and phrases seem faintly but undeniably to catch many others in the prism of their associations and connotations, almost as if Nabokov’s entire oeuvre were planned from the very start’ (viii). Turning to Pale Fire , the work of Simon Rowberry provides evidence of a whole network of ‘themed entries’ within this novel, what Rowberry refers to as ‘the novel’s promiscuous intertextuality.’ Alternately, the points and coordinates that Nabokov refers to constitute the composition’s ‘checked cells.’ The checked cells are the basic mechanism of the crossword puzzle; essentially, they are the guiding force of the entire puzzle, controlling both the construction and solution. These are the cells within the crossword puzzle in which two words intersect. In Nabokov’s compositional crossword, the ‘checked cells’ are those points which combine disparate entities, places of intersection, where objects and themes converge.”

Rowe, W.W., Nabokov’s Deceptive World , New York University Press, 1971.

Rowberry, Simon, “Pale Fire  as a Hypertextual Network.” 22nd ACM Hypertext Conf., Eindhoven, Netherlands. 6-9 June 2011. Web.

The Rowberry date appears to be, specifically, 8  June 2011:

A Kinbote note — See also this  journal on 8 June 2011.

Update of 3:03 PM ET the same day —

In keeping with Kinbote’s character as an unreliable narrator . . .
Rowberry’s Eindhoven slides  indicate he spoke on 9  June 2011.

See as well the Log24 post  “Historical Fiction” from June 2011.

Saturday, December 28, 2019

Caballo Blanco

Filed under: General — Tags: , , , — m759 @ 9:02 AM

The key  is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight

“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”

— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

“It must be remarked that these 8 heptads are the key  to an elegant proof….”

— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

Thursday, September 26, 2019

Identity

Filed under: General — Tags: , — m759 @ 8:53 AM


Related story:

E is for Einheit

Wednesday, September 25, 2019

Connecting the Dots

Filed under: General — Tags: , — m759 @ 5:27 PM

"A scholar in England suspected annotations in a First Folio
at the Free Library of Philadelphia were John Milton’s, so he
connected the dots . . . ." — The New York Times  last Thursday

It’s a combination of elation and fear, a certain kind of terror,”
Dr. Scott-Warren, a lecturer at Cambridge University, said
Thursday in an interview, describing his feelings.

“As a scholar, you get a sense of the fixed landmarks,” he said.
“Suddenly to have a new landmark to come right up through
the ground is quite disconcerting; there’s something alarming
about that.”
. . . .

"Dr. Scott-Warren  studies the history of books as material objects …."

Saturday, August 3, 2019

As:  A Four-Set

Filed under: General — Tags: , — m759 @ 11:04 AM

For further details, click on the image below.

Monday, May 27, 2019

But Seriously . . .

Filed under: General — Tags: , — m759 @ 9:32 PM

McLuhan on Analogy.

I prefer the simple "four dots" figure
of the double colon:

For those who prefer stranger analogies . . .

Actors from "The Eiger Sanction" —

Doctor Strange on Mount Everest —

Dr. Strange at beyondtheopposites.com on 2016/12/02

See as well this  journal on the above Strange date, 2016/12/02,
in posts tagged Lumber Room.

Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: — m759 @ 6:45 PM

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .

 .

Sunday, October 28, 2018

Commonwealth Tales, or “Lost in Physics”

Filed under: General — m759 @ 11:00 PM

From Ulysses , by James Joyce —

John Eglinton, frowning, said, waxing wroth:

—Upon my word it makes my blood boil to hear anyone compare Aristotle with Plato.

—Which of the two, Stephen asked, would have banished me from his commonwealth?

Compare and contrast:

Plato's diamond in Jowett's version of the Meno dialogue

Fans of Plato might enjoy tales of Narnia, but fans of
James Joyce and Edgar Allan Poe might prefer
a tale by Michael Chabon from April 2001 about a
"doleful little corner of western Pennsylvania."

Monday, April 23, 2018

Super Symmetry Surfing

Filed under: General — Tags: — m759 @ 11:17 AM

Midrash —

    

Backstory — Search this journal for Taormina.

Saturday, April 7, 2018

Dot

Filed under: General — Tags: — m759 @ 10:41 PM

The late Philip J. Davis in his 2004 essay 

"A Brief Look at Mathematics and Theology,"
Humanistic Mathematics Network Journal ,
Issue 27, Article 14. Available at:
http://scholarship.claremont.edu/hmnj/vol1/iss27/14/ 

wrote —

"In my childhood, the circle persisted as a potent magic figure
in the playtime doggerel 'Make a magic circle and sign it with a dot.'
The interested reader will find thousands of allusions to the phrase
'magic circle' on the Web."

There are fewer allusions to "magic circle" + "sign it with a dot."

One such allusion (click to enlarge) is . . .

Davis died on Pi Day .

Thursday, April 5, 2018

D8

Filed under: General — Tags: , , — m759 @ 10:00 PM

The above title may be regarded as a poetic variant
of the title of Katherine Neville's 1988 novel The Eight .

Related material —

See also The Black Queen, a note from 2001.

Wednesday, April 4, 2018

The Key

Filed under: General,Geometry — Tags: — m759 @ 12:32 PM

"The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG (3, 2) and Its Group," 
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in 
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference 
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

For those who, like the author of The Eight  (a novel in which today's
date figures prominently), prefer fiction —

See as well . . .

Literary theorists may, if they wish, connect
cabalistically the Insidious  address "414" 
with the date  4/14 of the above post, and
the word Appletree with the biblical Garden.

Monday, February 26, 2018

The Unfolding

Filed under: General,Geometry — Tags: — m759 @ 9:20 AM

From the website of Richard P. Gabriel

" As part of my studies, I came up with a 'theory of poetry' 
based loosely on Christopher Alexander’s 'Nature of Order.' "
[The Alexander link is mine, not Gabriel’s.]

A phrase from this  journal a year ago today — "poetic order" —
links to the theory of Gabriel —

From Gabriel's "The Nature of Poetic Order" —

Positive Space

• Positive space is the characteristic of a center
that moves outward from itself, seemingly oozing life
rather than collapsing on itself
• An image that resonates is showing positive space
• A word that has many connotations that fit with the
other centers in the poem is showing positive space
• It is an expansion outward rather than a contraction
inward, and it shows that the poem is unfolding
in front of us and not dying

Related material —

From a post of April 26, 2017

Tuesday, October 17, 2017

The Movement of Analogy

Filed under: General,Geometry — Tags: , — m759 @ 4:20 PM

The title is a phrase by Octavio Paz from today's post
"Status Symbols."

Other phrases from a link target in Sunday's post 
The Strength at the Centre

                                a single world
In which he is and as and is are one.

See also Four Dots in this journal.

Saturday, April 8, 2017

A Perfect Nonentity

Filed under: General,Geometry — Tags: , — m759 @ 8:25 AM

The title is from Hume:

 "And were all my perceptions removed by death,
and could I neither think, nor feel, nor see, nor love,
nor hate, after the dissolution of my body, I should
be entirely annihilated, nor do I conceive what is
further requisite to make me a perfect nonentity."

— Book I, Part IV, Section vi  of  
    A Treatise of Human Nature

"What is further requisite" — Perhaps  

This four-dot notation ("as") is from a search for Lévi-Strauss in this journal.

See also "That I Am."

Friday, December 2, 2016

Smoke from the Sacred Wood

Filed under: General — Tags: , — m759 @ 11:00 PM

The beginning of an essay by Emily Witt that is to appear on Sunday,
Dec. 4, 2016, in the T Magazine  of The New York Times

"Palo santo, which means 'holy stick' in Spanish, is a tree indigenous to the Caribbean and South America. When burned, it emits a fragrance of pine and citrus. Lighting a stick of palo santo, like burning a bundle of sage or sweetgrass, is believed to chase away misfortune. Amazonian shamans use it in ayahuasca ceremonies to cleanse a ceremonial space of bad spirits. Given its mystical connotations, it’s not a scent associated with the secular world, but lately I have noticed its distinctive smoke wafting over more earthly settings, from Brooklyn dive bars to blue-chip art openings."

The ending of an essay by T. S. Eliot that appeared in his 1921 book
titled The Sacred Wood

Those who prefer ayahuasca ceremonies may consult
a Sept. 10 post, Cocktail of the Damned.

Sunday, November 6, 2016

Fourplay

Filed under: General — Tags: — m759 @ 11:00 PM

On a renovation of Manhattan's Irish Repertory Theatre:

"Performances in the new space began on May 17, 2016."

This journal on May 17, 2016

Click the image below for a related story. 

See also Cartesian Theatre, a post of April 19, 2004.

Thursday, September 8, 2016

Afternoon Delight

Filed under: General — Tags: — m759 @ 3:26 PM

The notation "O" in the previous post suggests
a review of the new "Grease" theory in light of
a phrase from a May 2014 Oslo art exhibition

"a desperate sense of imagined community."

Illustration (click for a video) —

"I'll have what she's having."

See as well Olivia Newton-John in this journal as the Muse of Dance

June 29 Meditation

Filed under: General — m759 @ 6:29 AM

A Log24 noon post of June 29, 2016, The Mystery of O 
links to the following passage —

This use of "O" is not a notation I recommend.

Saturday, July 16, 2016

Binary Shema: The “O” and the “I”

Filed under: General,Geometry — Tags: — m759 @ 4:35 AM

The "I" of "I Am that I Am" has been described as a creation of
an "ur-unity" (see the Anderson passage below) and this ur-unity,
denoted by "O," has been described elsewhere as "a primary reality"
(see the Sullivan passage below). These descriptions are of course 
much less clear than those usually given for the similar purely
mathematical *  notations "0" and "1."

See also Quine's Shema  in "Is Nothing Sacred?" —

0! = 1.

Quoted here on July 30, 2015

Kabbalah and Finnegans Wake

Linked to here on June 29, 2016

Bion and the 'O'

*  Note for mathematicians: Here characteristic 0 is assumed .
    Quine's Shema does not apply to Galois.

Saturday, June 25, 2016

Season

Filed under: General — Tags: — m759 @ 3:00 PM

Detail —

 

See also a search in this journal for Notation .

Saturday, May 28, 2016

Plan 4

Filed under: General — Tags: , — m759 @ 2:45 AM

See a search in this journal for "As Is."

Tuesday, May 17, 2016

Bullshit Studies

Filed under: General,Geometry — Tags: , , , — m759 @ 1:06 AM

The originator of the phrase 'Fab Four' reportedly
died at 80 on Saturday, May 14, 2016.

This suggests a review of another noted four-set.

The above image is from a study of Lévi-Strauss's "Canonical Formula"

Midrash —

Log24 post titled 'As Is'

[Above photo of Lévi-Strauss and formula added June 6, 2016.]

Thursday, January 28, 2016

A6!

Filed under: General,Geometry — m759 @ 9:57 AM

The title refers to a line by Louis Menand quoted
at the end of the previous post.

There "a6!" refers to the chessboard square in
column a, row 6.  In Geometry of the I Ching,
this square represents Hexagram 61, "Inner Truth."

See also "inner truth" in this journal.

Monday, July 13, 2015

Block Designs Illustrated

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

The Fano Plane —

"A balanced incomplete block design , or BIBD
with parameters , , , , and λ  is an arrangement
of b  blocks, taken from a set of v  objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r  blocks, every block
contains exactly k  varieties, and every pair of
varieties appears together in exactly λ  blocks.
Such an arrangement is also called a
(, v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane] 
is a (7, 7, 3, 3, 1) design."

— Ezra Brown, "The Many Names of (7, 3, 1),"
     Mathematics Magazine , Vol. 75, No. 2, April 2002

W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ).  Cherowitzo has described,
without mentioning its close connection with the
Fano-plane design, the following —

"the (8,14,7,4,3)-design on the set
X = {1,2,3,4,5,6,7,8} with blocks:

{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."

We can arrange these 14 blocks in complementary pairs:

{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.

These pairs correspond to the seven natural slicings
of the following eightfold cube —

Another representation of these seven natural slicings —

The seven natural eightfold-cube slicings, by Steven H. Cullinane

These seven slicings represent the seven
planes through the origin in the vector
3-space over the two-element field GF(2).  
In a standard construction, these seven 
planes  provide one way of defining the
seven projective lines  of the Fano plane.

A more colorful illustration —

Block Design: The Seven Natural Slicings of the Eightfold Cube (by Steven H. Cullinane, July 12, 2015)

Sunday, May 3, 2015

Das Scheinen

Filed under: General — Tags: , — m759 @ 9:00 PM

The title of Saturday night's post, "Die Scheinung ," is taken from
a 1920 book on a German poet, where "Scheinung " is associated
with "Maja ," a German spelling of a word with the connotation of
"the veil of illusion."

The phrase "Das Scheinen " is closer to "The Shining" in the
novel of that title by Stephen King. Some related remarks —

From a review of Capobianco's Engaging Heidegger —

"refreshing for its clarity and scholarly precision"

Thursday, March 12, 2015

For Stephen King

Filed under: General — Tags: — m759 @ 3:12 PM

Doctor Steam

"Everybody's doin'
a brand new dance now…"

     "A corpse will be
transported by express!"
Under the Volcano

Wednesday, February 11, 2015

Dead Reckoning

Filed under: General — Tags: — m759 @ 5:28 PM

Continued from yesterday evening

IMAGE- Bogart in 'Casablanca' with chessboard

Today's mathematical birthday — 

Claude Chevalley, 11 Feb. 1909 – 28 June 1984.

From MacTutor —

Chevalley's daughter, Catherine Chevalley, wrote about
her father in "Claude Chevalley described by his daughter"
(1988):—

For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged.

The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons.

Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.

Monday, July 14, 2014

Hexagram 14

Filed under: General,Geometry — m759 @ 11:00 AM

Possession
in great measure

Related material:

Lead obituary in today’s online New York Times  and Los Angeles Times 

Maazel reportedly died on Sunday, July 13, 2014.

From a search in this journal for Iconic Notation,
a related image from August 14, 2010—

http://www.log24.com/log/pix10B/100814-DBsm.jpg

See also…

Epiphany

Geometry of the I Ching (Box Style)

Box-style I Ching , January 6, 1989

Friday, March 28, 2014

Chinese Rune

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

"The Geometry of the I Ching introduces something called the Cullinane sequence
for the hexagrams, and uses a notation based on the four sides and two diagonals
in a square to indicate the yin and yang lines. The resulting rune-like symbols
are intriguing…."

— Andreas Schöter's  I Ching  home page

Actually, the geometry is a bit deeper than the rune-like symbols.

" 'Harriet Burden has been really great to me,'
Rune says in an interview, 'not only as a collector
of my work but as a true supporter. And I think of her
as a muse for the project … ' "

— In The Blazing World , the artist known as Rune

Monday, February 10, 2014

Mystery Box III: Inside, Outside

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:28 PM

(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)

The Box

Inside the Box

Outside the Box

For the connection of the inside  notation to the outside  geometry,
see Desargues via Galois.

(For a related connection to curves  and surfaces  in the outside
geometry, see Hudson's classic Kummer's Quartic Surface  and
Rosenhain and Göpel Tetrads in PG(3,2).)

Saturday, October 19, 2013

Face, Voice, Table

Filed under: General — m759 @ 9:29 PM

  Et ignotas animum dimittit in artes
— Ovid, Metamorphoses , VIII, 188,
     epigraph to Joyce's Portrait

Log24 last month —

Paul Hertz, alias "Ignotus the Mage" —

"When we're doing the fortunetelling, as soon
as we finish capturing the face and the voice,
they get sent right over to the table." — Paul Hertz,
"Ignotus" video, 2013

Commentary:

"… ignotus  has faint connotations of lowness,
baseness, vulgarity"

— "International Eyesore: Joyce the Pornographer,"
by S. J. Boyd, pp. 31-60 in Troubled Histories, 
Troubled Fictions
 
, ed. by C. C. Barfoot et al.

Or not so faint.

Related material:

The villanelle from A Portrait , and
a Log24 post of St. Stephen's Day, 2011.

Sunday, June 30, 2013

Book Award

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM
 

"What on earth is
a 'concrete universal'?
"

— Said to be an annotation
(undated) by Robert M. Pirsig
of A History of Philosophy ,
by Frederick Copleston,
Society of Jesus.

In the spirit of the late Thomas Guinzburg

See also "Concrete Universal" in this journal.

Related material— From a Bloomsday reply
to a Diamond Theory  reader's comment, an excerpt—

The reader's comment suggests the following passages from
the book by Stirling quoted above—

 

Here Stirling plays a role analogous to that of Professor Irwin Corey
accepting the National Book Award for Gravity's Rainbow  in 1974.

Sunday, June 2, 2013

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

See the Klein correspondence  at SymOmega today and in this journal.

"The casual passerby may wonder about the name SymOmega.
This comes from the notation Sym(Ω) referring to the symmetric group
of all permutations of a set Ω, which is something all of us have
both written and read many times over."

Friday, February 1, 2013

Get Quotes

Filed under: General — Tags: , — m759 @ 4:01 PM

For Tony Kushner fans:

For logic fans:

IMAGE- NY Times market quotes, American Express Gold Card ad, Kevin Spacey in 'House of Cards' ad

John Searle on Derrida:

On necessity, possibility, and 'necessary possibility'

In the box-diamond notation, the axiom Searle quotes is

.

"The euclidean property guarantees the truth of this." — Wikipedia

Linking to Euclid

Clicking on "euclidean" above yields another Wikipedia article

"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."

Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.

Tuesday, November 13, 2012

Tiw’s Day

Filed under: General — m759 @ 9:48 AM

All our words from loose using have lost their edge.”
— Ernest Hemingway

A newly coined term— "Spyfall"—

— is good, but not as good as…

NORway.

(Click on NOR for some connotations.
 See also Norway in this journal.)

Tuesday, October 9, 2012

Too Much Meaning

Filed under: General — Tags: — m759 @ 12:00 PM

Last night’s post discussed ways of draining the world of meaning.

For some tastes, poets like Dante do the opposite, supplying too much  meaning.

See a New Republic  review, dated Oct. 5, in which Harvard atheist Helen Vendler discusses Dante’s

“… assertion that Beatrice herself  ‘was this number [nine],’ since nine is the square of three, the number belonging to the Trinity. Dante’s fantastic reasoning requires pages of annotation, which Frisardi, drawing on a number of commentators, furnishes to the bewildered reader. The theological elaboration of the number nine— merely one instance of how far from our own* are Dante’s habits of thought— will convince any doubting reader that the Vita Nuova  requires annotation far beyond what its pages might seem to demand.”

Related material— Ninefold in this journal, and remarks by Joseph Campbell in a post, Plan 9, from Sept. 5.

* Speak for yourself, Helen.

Thursday, June 7, 2012

The Field

Filed under: General,Geometry — m759 @ 2:56 AM

    "Time for you to see the field." —Bagger Vance

IMAGE- The Galois field GF(8) in binary and in algebraic notations

This post was suggested by a link from a post
 in this journal seven years ago yesterday—

Is the language of thought
 any more than a dream?

— Rimbaud

Yes.

Monday, January 30, 2012

Enda’s Game*

Filed under: General — Tags: — m759 @ 7:00 PM

The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review  that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")

"Lastly, tengwesta  [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe  (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe  only in great need and urgency, and especially when lambe  is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe  will use lambe  when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe  that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe  is also often taking place; for the will to converse in lambe  is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."

* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."

   — Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar  39 (July 1998). This is copyrighted but freely available on the Web.

** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is  such a thing as a tesseract."

Friday, January 6, 2012

Defining Form

Filed under: General,Geometry — Tags: , , — m759 @ 10:10 AM

IMAGE- MLA session, 'Defining Form,' chaired by Colleen Rosenfeld of Pomona College

Some related resources from Malcolm Lowry

"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie , Serpent and Siva Worship in Central America , there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King , probably they were treasures, but the rest were a heterogeneous collection…."

Under the Volcano , Chapter VI

— and from Matilde Marcolli

Seven books on analytical psychology

See also Marcolli in this morning's previous post, The Garden Path.

For the relevance of alchemy to form, see Alchemy in this journal.

Wednesday, September 7, 2011

The Most Important Configuration

Filed under: General,Geometry — m759 @ 6:30 PM

A search for some background on Gian-Carlo Rota's remarks
in Indiscrete Thoughts * on a geometric configuration
leads to the following passages in Hilbert and Cohn-Vossen's
classic Geometry and the Imagination

http://www.log24.com/log/pix11B/110907-HCV-BPconfigSm.jpg

These authors describe the Brianchon-Pascal configuration
of 9 points and 9 lines, with 3 points on each line
and 3 lines through each point, as being
"the most important configuration of all geometry."

Thus it seems worthwhile to relate it to the web page
on square configurations referenced here Tuesday.

The Encyclopaedia of Mathematics , ed. by Michiel Hazewinkel,
supplies a summary of the configuration apparently
derived from Hilbert and Cohn-Vossen

http://www.log24.com/log/pix11B/110907-HazewEnc-Brianchon-Pascal-Annot3Sm.jpg

My own annotation at right above shows one way to picture the
Brianchon-Pascal points and lines— regarded as those of a finite,
purely combinatorial , configuration— as subsets of the nine-point
square array discussed in Configurations and Squares. The
rearrangement of points in the square yields lines that are in
accord with those in the usual square picture of the 9-point
affine plane.

A more explicit picture—

http://www.log24.com/log/pix11B/110907-AG23lines500w.jpg

The Brianchon-Pascal configuration is better known as Pappus's  configuration,
and a search under that name will give an idea of its importance in geometry.

* Birkhäuser Boston, 1998 2nd printing, p. 145

Saturday, July 23, 2011

Death Argument

Filed under: General — Tags: — m759 @ 6:00 PM

Suggested by Peter J. Cameron's weblog post today on Halmos,
by a July 18 post in this  journal on the Norwegian mathematician Abel,
by a link in the July 18 post to  "Death Proof," and by today's
midday New York Lottery (459 and 7404)—

From July 4, 2004 (7404 interpreted as a date)—

"There are two unfortunate connotations of 'proof' that come from mathe-
matics and make the word inappropriate in discussions of the security of cryp-
tographic systems. The first is the notion of 100% certainty. Most people not
working in a given specialty regard a 'theorem' that is 'proved' as something
that they should accept without question. The second connotation is of an intri-
cate, highly technical sequence of steps. From a psychological and sociological
point of view, a 'proof of a theorem' is an intimidating notion: it is something
that no one outside an elite of narrow specialists is likely to understand in detail
or raise doubts about. That is, a 'proof' is something that a non-specialist does
not expect to really have to read and think about.

The word 'argument,' which we prefer here, has very different connotations."

— "Another Look at 'Provable Security'," 
by Neal Koblitz and Alfred J. Menezes, July 4, 2004
(updated on July 16, 2004; October 25, 2004; March 31, 2005; and May 4, 2005)

As for 459, see Post  459 in this journal.

Related material: The Race, Crossing the Bridge, Aristophanic View, and Story Theory.

Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: , — m759 @ 9:29 AM

Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest  thing there is
(Tractatus Logico-Philosophicus  No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach  sein. Sie ist vor  aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh.  No. 5.5563.)

See also

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

* Updates of 9:29 PM ET July 10, 2011—

A  mnemonic  from a course titled “Galois Connections and Modal Logics“—

“Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and diamond (diamond) for possibly. Then, for example, diamondP  can be read as
‘it is possibly the case that P .'”

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

“The diamond symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’”

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

Tuesday, May 17, 2011

Cannes Bangs

Filed under: General — m759 @ 5:55 PM

24 Frames
MOVIES: PAST, PRESENT AND FUTURE (LA Times )

"But what feels like standard movie exposition quickly takes
a sharp turn when we're feted with about 20 minutes of the
elemental and cosmic footage that's been making all the
headlines. At first it looks like it could be a depiction of heaven
or hell, but it soon becomes clear that it's a story of creation—
or of Creation, as some iteration of the Big Bang unfolds
before our eyes."

— "Cannes 2011: What Terrence Malick's 'The Tree of Life'
Is Actually About," by Steven Zeitchik of the LA Times

Hannibal Pictures
THE BIG BANG (Click for Cannes details.)

http://www.log24.com/log/pix11A/110517-HannibalPicturesHands.jpg

See Peter Woit's review from Sunday.

The generic 3×3 HannibalPictures.com
favicon  has an apt connotation

Plan 9 from Outer Space.

Friday, April 15, 2011

Spider Notes

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

http://www.log24.com/log/pix11/110415-Symm-axes.jpg

Some connotations of the word "eightfold" —

IMAGE- Google search for 'eightfold geometry,' April 15, 2011

See also Damnation Morning and today's New York Times

A Final Bow for Julie Taymor's 'Spider-Man' Vision.

Thursday, September 23, 2010

Evocation*

Filed under: General — m759 @ 6:00 AM

"Epistulae ad familiares" (adfamiliares for short) at livejournal.com

"Prefatory notes evoke a Republic of Letters— or at least an academic support group— in which the writer claims membership. In fact, they often describe something much more tenuous, the group of those who the author wishes had read his work, offered him references, or at least given him the time of day. Hence they retain something of the literary— not to say fictional— quality of traditional poets' prayers." (Anthony Grafton, The Footnote: A Curious History)

P.S. This book rules.  Why did I wait so long to read it?

* See a definition. See also this  journal's previous post, Patterns in the Carpets. As for "those who the author wishes had read his work," see a quotation from an author mentioned in that post, Greg Egan, that seems relevant to the suicide outside Harvard's Memorial Church last Saturday during the morning Yom Kippur service—

… The word "transhumanism" (or, even worse, "posthumanism") sounds like a suicide note for the species, which effectively renders it a political suicide note for any movement by that name. No doubt there are people prepared to spend 90% of their time and energy explaining that they didn't intend  any negative connotations, but this is not one of those cases where other people will be to blame if "transhumanists" are reviled as the enemies of humanity on purely linguistic grounds. It's no use people proclaiming "Please, read my 1,000-page manifesto, don't just look at one word!"….

— Greg Egan on April 23, 2008,** at Metamagician and the Hellfire Club

Related material— A livejournal note on the Memorial Church suicide, nihilism, and a "final crux."

** Footnote to a footnote— See also Log24 on April 23, 2008— Shakespeare's birthday.

Monday, August 16, 2010

Utopia 14

Filed under: General — Tags: , — m759 @ 4:16 AM

The following, from Wikipedia, is an image of Utopia 14, the 1954 paperback reissue of Kurt Vonnegut’s 1952 novel Player Piano.

http://www.log24.com/log/pix10B/100816-Utopia14-Vonnegut.jpg

Commentary from Wikipedia

“A player piano is a modified piano that ‘plays itself.’ The piano keys move according to a pattern of holes punched in an unwinding scroll…. Like its counterpart, a player piano can be played by hand as well. When a scroll is run through the ghost-operated instrument, the movement of its keys produce the illusion that an invisible performer is playing the instrument.”

See also last night’s “The Game“—

“One would call out, in the standardized abbreviations of their science, motifs or initial bars of classical compositions, whereupon the other had to respond with the continuation of the piece, or better still with a higher or lower voice, a contrasting theme, and so forth. It was an exercise in memory and improvisation….”

— as well as Vonnegut in this journal yesterday and the following from the August 14 post Iconic Notation

A question from Ivan Illich
(founder of CIDOC, the Center for Intercultural Documentation,
in Cuernavaca, Mexico)—

Who can be served by bridges to nowhere?

For more about nowhere, see Utopia.

For more about Cuernavaca and ghosts, see a recurring motif in this journal.

Monday, June 14, 2010

Birkhoff on the Galois “Theory of Ambiguity”

Filed under: General,Geometry — m759 @ 9:48 PM

The Principle of Sufficient Reason

by George David Birkhoff

from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940

EXCERPT 1—

My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2

In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is

2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought

3 That is, uses multi-dimensional symbols beyond our grasp.
______________________________________________________________________

the appropriate instrument of thought to remedy our deficiency
in this respect.

The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.5

Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.

Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D;  B, C, D, A;  C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a

4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________

null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:

http://www.log24.com/log/pix10A/100614-BirkhoffTable.jpg
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________

EXCERPT 2—

Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:

PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).

HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.

The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.

In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.

It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:

Image-- Birkhoff diagram relating Galois's theory of ambiguity to metaphysics

While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.

___________________________________________________________________________

A more recent lecture on the same subject —

"From Leibniz to Quantum World:
Symmetries, Principle of Sufficient Reason
and Ambiguity in the Sense of Galois
"

by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)

Sunday, June 13, 2010

Sunday School

Filed under: General — m759 @ 9:00 AM

What on earth is a 'concrete universal'?"
Said to be an annotation (undated)
by Robert M. Pirsig of A History of Philosophy,
by Frederick Copleston, Society of Jesus.

From Aaron Urbanczyk's 2005 review of Christ and Apollo  by William Lynch, S.J., a book first published in 1960—

"Lynch's use of analogy vis-a-vis literature provides, in a sense, a philosophical basis to the theoretical paradox popularized by W. K. Wimsatt (1907-1975), which contends that literature is a sort of 'concrete universal.'"

The following figure has often been
offered in this journal as a symbol of Apollo

Image-- 3x3 array of white squares

Arguments that it is, rather, a symbol of Christ
may be left to the Society of Jesus.

One possible approach—
Urbanczyk's review says that
"Christianity offers the critic
   a privileged ontological window…."

"The world was warm and white when I was born:
Beyond the windowpane the world was white,
A glaring whiteness in a leaded frame,
Yet warm as in the hearth and heart of light."

Delmore Schwartz

Wednesday, February 3, 2010

Holly Day

Filed under: General — Tags: — m759 @ 6:00 AM

Today's word:

The musical notation 'fermata,' or 'birdseye'

fermata

"February made me shiver…."
American Pie

Tuesday, October 13, 2009

Tuesday October 13, 2009

Filed under: General — m759 @ 7:00 AM
Wakes


This morning’s New York Times
reports the deaths of Nuremberg interrogator Richard W. Sonnenfeldt and of avant-garde novelist and Beckett scholar Raymond Federman.

Symbols from this journal on the dates of their deaths:

For Sonnenfeldt, who died
 on Friday, Oct. 9,
a symbol from that date:

The 3x3 grid as religious symbol


For connotations of the symbol appropriate to the name Sonnenfeldt, see the link to A Sunrise for Sunrise in the entry of Saturday, Oct. 10.

For Federman, who died
 on Tuesday, Oct. 6,
a symbol from that date:

Black monolith

A quotation that appeared here on Wednesday, Oct. 7, seems relevant to Federman:

But I am a worker, a tombstone mason, anxious to pleace averyburies and jully glad when Christmas comes his once ayear. You are a poorjoist, unctuous to polise nopebobbies….

— James Joyce, Finnegans Wake

Thursday, July 2, 2009

Thursday July 2, 2009

Filed under: General — Tags: , — m759 @ 12:00 PM

Hieron Grammaton, Part III*

The Old Man and the Light

In memory of
Ernest Hemingway,
who died on this date
in 1961, a story
in three parts:

 

I — Eye of Raven

The musical notation 'fermata,' or 'birdseye'

Fermata

 

II — Psyche and Symbol

Leonard Baskin, detail of cover of Jung's 'Psyche and Symbol'

Leonard Baskin, detail of
cover for Jung’s
Psyche and Symbol

 

III — Raven Steals the Light

The box of light from animated video of 'Raven Steals the Light'

Detail from the story
Raven Steals the Light

Midrash:

“To the earnestness of death belongs precisely that capacity for awakening, that resonance of a profound mockery which, detached from the thought of the eternal, is an empty and often brash jest, but together with the thought of the eternal is just what it should be….” —Kierkegaard

* For Hieron Grammaton, Parts I and II, see the five Log24 entries from 6:29 PM Tuesday, June 23, to 1:00 AM Sunday, June 28.

Sunday, April 12, 2009

Sunday April 12, 2009

Filed under: General — Tags: — m759 @ 3:09 AM
Where Entertainment
Is God
, continued

Dialogue from the classic film Forbidden Planet

"… Which makes it a gilt-edged priority that one of us gets into that Krell lab and takes that brain boost."

— Taken from a video (5:18-5:24 of 6:09) at David Lavery's weblog in the entry of Tuesday, April 7.

(Cf. this journal on that date.)

Thanks to Professor Lavery for his detailed notes on his viewing experiences.

My own viewing recently included, on the night of Good Friday, April 10, the spiritually significant film Indiana Jones and the Kingdom of the Crystal Skull.

The mystic circle of 13 aliens at the end of that film, together with Leslie Nielsen's Forbidden Planet remark quoted above, suggests the following:

"The aim of Conway’s game M13 is to get the hole at the top point and all counters in order 1,2,…,12 when moving clockwise along the circle." —Lieven Le Bruyn

 

http://www.log24.com/log/pix09/090411-M13.gif

The illustration is from the weblog entry by Lieven Le Bruyn quoted below. The colored circles represent 12 of the 13 projective points described below, the 13 radial strokes represent the 13 projective lines, and the straight lines in the picture, including those that form the circle, describe which projective points are incident with which projective lines. The dot at top represents the "hole."

From "The Mathieu Group M12 and Conway’s M13-Game" (pdf), senior honors thesis in mathematics by Jeremy L. Martin under the supervision of Professor Noam D. Elkies, Harvard University, April 1, 1996–

"Let P3 denote the projective plane of order 3. The standard construction of P3 is to remove the zero point from a three-dimensional vector space over the field F3 and then identify each point x with -x, obtaining a space with (33 – 1)/2 = 13 points. However, we will be concerned only with the geometric properties of the projective plane. The 13 points of P3 are organized into 13 lines, each line containing four points. Every point lies on four lines, any two points lie together on a unique line, and any two lines intersect at a unique point….

Conway [3] proposed the following game…. Place twelve numbered counters on the points… of P3 and leave the thirteenth point… blank. (The empty point will be referred to throughout as the "hole.") Let the location of the hole be p; then a primitive move of the game consists of selecting one of the lines containing the hole, say {p, q, r, s}. Move the counter on q to p (thus moving the hole to q), then interchange the counters on r and s….

There is an obvious characterization of a move as a permutation in S13, operating on the points of P3. By limiting our consideration to only those moves which return the hole to its starting point…. we obtain the Conway game group. This group, which we shall denote by GC, is a subgroup of the symmetric group S12 of permutations of the twelve points…, and the group operation of GC is concatenation of paths. Conway [3] stated, but did not prove explicitly, that GC is isomorphic to the Mathieu group M12. We shall subsequently verify this isomorphism.

The set of all moves (including those not fixing the hole) is given the name M13 by Conway. It is important that M13 is not a group…."

[3] John H. Conway, "Graphs and Groups and M13," Notes from New York Graph Theory Day XIV (1987), pp. 18–29.


Another exposition (adapted to Martin's notation) by Lieven le Bruyn (see illustration above):

 

"Conway’s puzzle M13 involves the 13 points and 13 lines of P3. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the 'hole' (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture [above] the lines are represented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway's M13 puzzle having the hole at the top position contains the sporadic simple Mathieu group M12 as a subgroup."

For the religious significance of the circle of 13 (and the "hole"), consider Arthur and the 12 knights of the round table, et cetera.

But seriously…
 
Delmore Schwartz, 'Starlight Like Intuition Pierced the Twelve'

Sunday, March 1, 2009

Sunday March 1, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

Solomon's Cube
continued

"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."

— G. H. Hardy, A Mathematician's Apology

"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia

"One has O+(6) ≅ S8, the symmetric group of order 8! …."

 — "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.

"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 AM
Wechsler Cubes
“Confusion is nothing new.”
— Song lyric, Cyndi Lauper
Part I:
Magister Ludi

Hermann Hesse’s 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–

“For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar’s Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors.”

Part II:
A Bulky Memorandum

From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–

A description of the work of Bill Wechsler, a fictional artist:

“Bill worked long hours on a series of autonomous pieces about numbers. Like O’s Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has

— End of page 168 —

opened the door to his cell. The hidden rhyme is “free.” Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn’t visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe’s color wheel and in Alfred Jensen’s use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn’t bother with the poet’s meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.

The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page.”

Part III:
Wechsler Cubes(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) —

From 2002:

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

Part IV:
A Magic Gallery
Log24, March 4, 2004

ZZ
WW

Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:


Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte

Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen

 Numbers and Names,
Wording and Words

With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes


English translation
by Catherine Schelbert
A related poem:

Alphabets
by Hermann Hesse

From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone’s command;
it is a game whose rules
are nice and neat.

But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens’
feet upon the snow.
He’d hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He’d see the way love burns
and anguish trembles,
He’d wonder, laugh,
shake with fear and weep
because beyond this cipher’s
cross-barred keep
he’d see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
prisoner-fashion.
He’d think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He’ll sigh, and smile,
and feel all right again.

— Hermann Hesse (1943),
Buchstaben,” from
Das Glasperlenspiel,
translated by
Richard and Clara Winston

Tuesday, February 26, 2008

Tuesday February 26, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

Eight is a Gate (continued)

Tom Stoppard, Jumpers:
"Heaven, how can I believe in Heaven?" she sings at the finale. "Just a lying rhyme for seven!"
"To begin at the beginning: Is God?…" [very long pause]

 
From "Space," by Salomon Bochner

Makom. Our term “space” derives from the Latin, and is thus relatively late. The nearest to it among earlier terms in the West are the Hebrew makom and the Greek topos (τόπος). The literal meaning of these two terms is the same, namely “place,” and even the scope of connotations is virtually the same (Theol. Wörterbuch…, 1966). Either term denotes: area, region, province; the room occupied by a person or an object, or by a community of persons or arrangements of objects. But by first occurrences in extant sources, makom seems to be the earlier term and concept. Apparently, topos is attested for the first time in the early fifth century B.C., in plays of Aeschylus and fragments of Parmenides, and its meaning there is a rather literal one, even in Parmenides. Now, the Hebrew book Job is more or less contemporary with these Greek sources, but in chapter 16:18 occurs in a rather figurative sense:

O earth, cover not thou my blood, and let my cry have no place (makom).

Late antiquity was already debating whether this makom is meant to be a “hiding place” or a “resting place” (Dhorme, p. 217), and there have even been suggestions that it might have the logical meaning of “occasion,” “opportunity.” Long before it appears in Job, makom occurs in the very first chapter of Genesis, in:

And God said, Let the waters under the heaven be gathered together unto one place (makom) and the dry land appear, and it was so (Genesis 1:9).

This biblical account is more or less contemporary with Hesiod's Theogony, but the makom of the biblical account has a cosmological nuance as no corresponding term in Hesiod. Elsewhere in Genesis (for instance, 22:3; 28:11; 28:19), makom usually refers to a place of cultic significance, where God might be worshipped, eventually if not immediately. Similarly, in the Arabic language, which however has been a written one only since the seventh century A.D., the term makām designates the place of a saint or of a holy tomb (Jammer, p. 27). In post-biblical Hebrew and Aramaic, in the first centuries A.D., makom became a theological synonym for God, as expressed in the Talmudic sayings: “He is the place of His world,” and “His world is His place” (Jammer, p. 26). Pagan Hellenism of the same era did not identify God with place, not noticeably so; except that the One (τὸ ἕν) of Plotinus (third century A.D.) was conceived as something very comprehensive (see for instance J. M. Rist, pp. 21-27) and thus may have been intended to subsume God and place, among other concepts. In the much older One of Parmenides (early fifth century B.C.), from which the Plotinian One ultimately descended, the theological aspect was only faintly discernible. But the spatial aspect was clearly visible, even emphasized (Diels, frag. 8, lines 42-49).

BIBLIOGRAPHY

Paul Dhorme, Le livre de Job (Paris, 1926).

H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed. (Berlin, 1938).

Max Jammer, Concepts of Space (Cambridge, Mass., 1954).

J. M. Rist, Plotinus: The Road to Reality (Cambridge, 1967).

Theologisches Wörterbuch zum Neuen Testament (1966), 8, 187-208, esp. 199ff.

— SALOMON BOCHNER

Related material: In the previous entry — "Father Clark seizes at one place (page eight)
upon the fact that…."

Father Clark's reviewer (previous entry) called a remark by Father Clark "far fetched."
This use of "place" by the reviewer is, one might say, "near fetched."

Saturday, August 18, 2007

Saturday August 18, 2007

Filed under: General — m759 @ 7:20 PM
A Concrete Universal

What on earth is
a ‘concrete universal’?

Said to be an annotation
(undated) by Robert M. Pirsig
of A History of Philosophy,
by Frederick Copleston,
Society of Jesus.

No matter how it’s done,
you won’t like it.

— Robert Redford to     
  Robert M. Pirsig in Lila    


“In chapters 19 and 20 of LILA there is a discussion about the possibility of making Zen and the Art into a movie. It opens with a scene where Robert Redford, who ‘really would like to have the film rights,’ comes to meet and negotiate with Phaedrus in his New York City hotel room. Phaedrus tells the famous actor that he can have the rights to the book, but maybe that’s just because he’s star-struck and doesn’t like to haggle. Under his excitement, Phaedrus has a bad feeling about it. He tells us that he’s been warned by several different people not to allow such a film to be made. Even Redford warned him not to do it. So what’s the problem? As it’s put at the end of that discussion, ‘Films are social media; his book was largely intellectual. That was the center of the problem.'”

David Buchanan at robertpirsig.org

“The insight is constituted precisely by ‘seeing’ the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete.”

— Fr. Brian Cronin‘s Foundations of Philosophy, Ch. 2, “Identifying Direct Insights,” quoted in Ideas and Art

See also Smiles of a Summer Evening, the current issue of TIME, the time of this entry (7:20:11 PM ET), and Plato, Pegasus, and the Evening Star.

Thursday, March 22, 2007

Thursday March 22, 2007

Filed under: General — m759 @ 12:00 PM
Chess Letter:
x

Queen sacrifice

Click on a picture
for the meaning of
the chess notation.
 
“Shakespeare, Rilke, Joyce,
Beckett and Levi-Strauss are
instances of authors for whom
chiasmus and chiastic thinking
are of central importance,
for whom chiasmus is a
generator of meaning,
tool of discovery and
  philosophical template.”
 
— Chiasmus in the Drama of Life

Monday, November 27, 2006

Monday November 27, 2006

Filed under: General — m759 @ 2:45 AM
The Poetry
of Philosophy

“What on earth is
   a ‘concrete universal’?”

Said to be an annotation
(undated)
by Robert M. Pirsig of
A History of Philosophy,
by Frederick Copleston,
Society of Jesus
.

For an answer, see
The Structure of the
‘Concrete Universal’
in Literature
,”
by W. K. Wimsatt, Jr.,
PMLA, Vol. 62, No. 1
(March, 1947), pp. 262-280.

This is reprinted in Wimsatt’s
The Verbal Icon:
Studies in the
Meaning of Poetry
.

The final chapter of
The Verbal Icon
is titled
“Poetry and Christian Thinking.”
For more on Wimsatt
and this topic, see
Reclaiming the Bible
as Literature,”
by Louis A. Markos.

Tuesday, October 10, 2006

Tuesday October 10, 2006

Filed under: General — Tags: , — m759 @ 8:00 PM
Mate in
Two Seconds

From Oct. 14 last year:

The image “http://www.log24.com/log/pix05B/051014-Tick.gif” cannot be displayed, because it contains errors.

From Oct. 13 last year
(Yom Kippur):

A Poem for Pinter
Oct. 13, 2005

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth.  The words 'necessary' and 'possible' are called modal connectives, or modalities.  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.  (For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.

That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.

For more on the former, see
Death in Two Seconds.

For more on the latter, see
The Halmos Tombstone.

4x9 black monolith

Thursday, October 13, 2005

Thursday October 13, 2005

Filed under: General — Tags: , , — m759 @ 12:00 PM

A Poem for Pinter

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise  (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary  and possible  truth.  The words 'necessary' and 'possible' are called modal connectives , or modalities .  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.

(For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: General,Geometry — m759 @ 8:16 AM

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

Friday, July 1, 2005

Friday July 1, 2005

Filed under: General — m759 @ 12:00 AM

Shining Through

From Dogma

“You see, Malloy, I’m writing a novel about Los Angeles…. It’s a fantastic place, you know, Malloy…. It has a Spanish name, with religious Roman Catholic connotations….”

From timesonline.co.uk, quotes of the day on May 19, 2005:

“My granddaughter once said I have a big imagination. And I said, ‘What’s a big imagination?,’ and she said, ‘You remember what never happened.'”

Isabel Allende, novelist, whose new book is based on the life of Zorro

“You all know I love LA, but tonight I really love LA.”

Antonio Villaraigosa, voted in as the city’s first Hispanic mayor in more than a century, thanks voters

See also
  Log24 entries ending at midnight
  August 28, 2003, and
  Log24 entries ending at midnight
  May 19, 2005,
  as well as the following illustrations
  from a Monday entry and
  from the entry it links to:

 Dream of Heaven


  (See also 3/3/04 and
             10/27/03.) 

The image “http://www.log24.com/log/pix05A/050701-ZZ.jpg” cannot be displayed, because it contains errors.

Tuesday, March 15, 2005

Tuesday March 15, 2005

Filed under: General — m759 @ 2:56 PM

Religion at Harvard

The Children’s Hour

Harvard Magazine,
Sept.-Oct. 2004
:

The image “http://www.log24.com/log/pix05/050315-Masters.jpg” cannot be displayed, because it contains errors.

“With the legalization of gay marriage in Massachusetts, Harvard couples were among those who took vows…. Lowell House master Diana Eck (left) and co-master Dorothy Austin tied the knot in Memorial Church on July 4, with Rev. Peter Gomes, Plummer professor of Christian morals, officiating.”

Once in Love with Amy

The image “http://www.log24.com/log/pix05/050315-Amy.jpg” cannot be displayed, because it contains errors.

Harvard’s
Lowell House:

“In the Dining Hall are portraits of President Lowell and his wife; his sister Amy Lowell (Pulitzer prize winning poet, and a lover of scandal…)….”

The image “http://www.log24.com/log/pix05/050315-Sharon.jpg” cannot be displayed, because it contains errors.

Today’s
Harvard Crimson:

“Stone joined members of the Foundation for lunch yesterday in Lowell House before delivering her remarks at Memorial Church last night…”


Hold That Thought

The image “http://www.log24.com/log/pix05/050315-Basic2.jpg” cannot be displayed, because it contains errors.

nothing – the word had sexual connotations, as a slang word referring to female sexual parts. Compare Hamlet:

HAMLET   Lady, shall I lie in your lap?
[Lying down at OPHELIA’s feet]
OPHELIA  No, my lord.
HAMLET   I mean, my head upon your lap?
OPHELIA  Ay, my lord.
HAMLET   Do you think I meant country matters?
OPHELIA  I think nothing, my lord.
HAMLET   That’s a fair thought to lie between maids’ legs.
OPHELIA  What is, my lord?
HAMLET   Nothing.

— Hamlet, III.2

Sunday, April 11, 2004

Sunday April 11, 2004

Filed under: General,Geometry — m759 @ 3:28 PM

Good Friday and
Descartes’s Easter Egg

“The use of z, y, x . . . to represent unknowns is due to René Descartes, in his La géometrie (1637)…. In a paper on Cartesian ovals, prepared before 1629, x alone occurs as unknown…. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown.”

— Florian Cajori, A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929. (Vol. 1, page 381)

This is from

http://members.aol.com/jeff570/variables.html.

Descartes’s Easter Egg is found at

EggMath: The Shape of an Egg —
Cartesian Ovals
 

An Easter Meditation
on Humpty Dumpty

The following is excerpted from a web page headed “Catholic Way.”  It is one of a series of vicious and stupid Roman Catholic attacks on Descartes.  Such attacks have been encouraged by the present Pope, who today said “may the culture of life and love render vain the logic of death.”

The culture of life and love is that of the geometry (if not the philosophy) of Descartes.  The logic of death is that of Karol Wojtyla, as was made very clear in the past century by the National Socialist Party, which had its roots in Roman Catholicism.

Humpty Dumpty sat on a wall.
Humpty Dumpty had a great fall

“In the century just completed, the human race found itself in a position not unlike the scrambled mess at the base of an imaginary English wall….

… we are heirs to a humanity that is broken, fractured, confused, unsure of what to make of itself….

 … ‘postmodernism’ is merely the articulation of the fractured, dissipated state of the human being…. 

Without relating a history of modern philosophy, our unfortunate human shell has suffered a continual fragmentation for a period of roughly 500 years. (You philosophers out there will recognize immediately that I am referring to the legacy of René Descartes.) And this fragmentation has been a one-way street: one assault after another on the integrity and dignity of the human person until you have, well, the 20th Century.

But now it’s the 21st Century.

The beauty … the marvel … the miracle of our time is the possibility that gravity will reverse itself: Humpty Dumpty may be able, once again, to assume his perch.”

—  Ted Papa,
Raising Humpty Dumpty

Voilà.

The upper part
of the above icon
is from EggMath.
For the lower part,
see Good Friday.

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.

Friday, July 11, 2003

Friday July 11, 2003

Filed under: General — m759 @ 10:23 PM

Father, Son,
and Holy Coast

Here are some religious meditations for the holy day 7-11:

As the website Hollywood Jesus perceptively points out, defending the story theory of truth, “Images that carry universal truths move us from the mundane to the sacred.  Jesus knew this when he spoke in parables.”

Here is a parable about my own name.

The Hollywood Jesus site tries to connect the cross of Christ, “holy wood,” with Hollywood by claiming that the words “holly” and “holy” are cognate.

See Hollywood and the Cross.

From the Online Etymology Dictionary

holly – O.E. holegn, from P.Gmc. *khuli-.

holy – O.E. halig “holy,” from P.Gmc. *khailagas. Adopted at conversion for L. sanctus. Primary meaning may have been “that must be preserved whole or intact, that cannot be transgressed or violated,” which would connect it with O.E. hal (see whole).

This shows that the holly-holy connection is, pace Neil Diamond, like nearly every other Christian claim, a damned lie.

Connoisseurs of junk culture may enjoy
a midi of Neil Diamond as background
for this Hollywood Jesus.

Here is a different Hollywood etymology that may be somewhat truer.

From the RootsWeb.com archives:

Re: CULLINANE-HOLLYWOOD-holly tree

“Cullen in Irish is Ó Cuillin (holly tree). …  This astonishingly simple name has worked its way through an astonishing number of variations including Cullion, Culhoun, MacCullen and Cullinane. …

In a message dated 6/5/01 8:24:18 PM Pacific Daylight Time, lawlerc@aol.comnojunk writes:

‘I do not have the surname in my family, but while looking at the Old Age Pension applications for the Barony of Strabane Upper, in the County of Tyrone, there was a notation that

the English equivalent of the surname CULLINANE is HOLLYWOOD.‘ “

Tuesday, June 24, 2003

Tuesday June 24, 2003

Filed under: General — m759 @ 7:59 PM

In memory of Leon Uris:

Mate Change Problem

White to mate in 2 moves

H.W.Grant, 1st Prize,
Australian Column 1924

The concept of “mate change,” appropriate on this, the coronation date of Henry VIII, is explained at Chathurangam.com, my source for the above problem.

For the connection with Leon Uris, find the “key” to the above chess problem… i.e., the notation for White’s first move.

From the New York Times, June 24:

“Reviewing Mr. Uris’s 1976 novel Trinity in The New York Times Book Review, Pete Hamill wrote: ‘Leon Uris is a storyteller, in a direct line from those men who sat around fires in the days before history and made the tribe more human.'”

Uris, 78, died at the summer solstice… Saturday, June 21, 2003. 

See also Force Field of Dreams.

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — m759 @ 9:26 PM

 STAR WARS  
opened on this date in 1977.

From the web page Amande:

Le Christ et la Vierge apparurent souvent entourés d’une auréole en forme d’amande: la mandorle.

Étymologiquement, le mot amande est une altération de amandala, qui dérive lui-même du latin classique amygdala….

L’amande a… une connotation symbolique, celle du sexe féminin. Elle figure souvent la vulve. Elle est alors en analogie avec la yoni du vocabulaire de l’hindouisme, la vulve ou la matrice, représentée par une amande ou une noix coupée en deux.

Screenshot of the online
New York Times, May 25, 2003:

Ariel the Hutt and Princess Amygdala

Introduction to Yantra

by Horia Cristescu and
Dan Bozaru 

The Triangle (TRIKONA)
The triangle (TRIKONA) is the symbol of
SHAKTI , the feminine energy or aspect of Creation. The triangle pointing down represents the YONI , the feminine sexual organ and the symbol of the supreme source of the Universe, and when the triangle is pointing upwards it signifies intense spiritual aspiration, the sublimation of one’s nature into the most subtle planes and the element of fire (AGNI TATTVA). The fire is always oriented upwards, thus the correlation with the upward triangle – SHIVA KONA. On the other hand, the downward pointing triangle signifies the element of water which always tends to flown and occupy the lowest possible position. This triangle is known as SHAKTI KONA.

The intersection of two geometric forms (lines, triangles, circles, etc.) represents forces that are even more intense than those generated by the simple forms. Such an interpenetration indicates a high level in the dynamic interaction of the correspondent energies. The empty spaces generated by such combinations are described as very efficient operational fields of the forces emanating from the central point of the YANTRA. That is why we can very often encounter representations of MANTRAS in such spaces. YANTRA and MANTRA are complementary aspects of SHIVA and their use together is much more efficient than the use of one alone.


The Six Points Star (SHATKONA)
A typical combination often found in the graphical structure of a YANTRA is the superposition of two triangles, one pointing upwards and the other downwards, forming a star with six points (SHATKONA), also known as David’s Star. This form symbolically represents the union of
PURUSHA and PRAKRITI or SHIVA-SHAKTI, without which there could be no Creation.

AMEN.

Sunday, March 2, 2003

Sunday March 2, 2003

Filed under: General — Tags: — m759 @ 10:20 PM

7:20 PM CALI Time

The Bus and the Bead Game:
 
The Communion of Saints as
 the Association of Ideas

On this date in 1955, “Bus Stop,” a play by William Inge, opened at the Music Box Theatre in New York City.

“I seemed to be standing in a bus queue by the side of a long, mean street.”

— C. S. Lewis, The Great Divorce, opening sentence

Today’s birthdays:

Sam Houston
Dr. Seuss
Kurt Weill
Mikhail Gorbachev
Tom Wolfe
Desi Arnaz
Jennifer Jones
Karen Carpenter

and many others.

Today is the feast day of  

St. Randolph Scott
St. Sandy Dennis
St. D. H. Lawrence, and
St. Charlie Christian.

“Your guitar, it sounds so sweet and clear…”

— Karen Carpenter singing “Superstar

“And if I find me a good man,
 I won’t be back at all.”

C. C. Rider lyrics

See (and hear) also “Seven Come Eleven,” played by St. Charlie Christian.

One might (disregarding separation in time and space — never major hindrances to the saints) imagine C. S. Lewis in Heaven listening to a conversation among the four saints listed above.  For more on the communion of saints, see my entry “State of the Communion” of Tuesday, Jan. 28, 2003.  This entry, quoting an old spiritual, concluded with “Now hear the word of the Lord”  — followed by this notation

 7:11 PM.

See also the N.Y. Times obituary of John P. Thompson of Dallas, former 7-Eleven chairman, who died, as it happened, on that very day (Jan. 28).  See also Karen Carpenter’s “first take luck.”

The sort of association of ideas described in the “Communion” entry is not unrelated to the Glasperlenspiel, or Glass Bead Game, of Hermann Hesse.  For a somewhat different approach to the Game, see

The Glass Bead Game,”

by John S. Wilson, group theorist and head of the Pure Mathematics Group at the University of Birmingham in England. Wilson is “not convinced that Hesse’s… game is only a metaphor.” Neither am I.

For the association-of-ideas approach, see the page cited in my “Communion” entry,

A Game Designer’s Holy Grail,”

and (if you can find a copy) one of the greatest forgotten books of the twentieth century,

The Third Word War,

by Ian Lee (A&W Publishers, Inc., New York, 1978).  As Lee remarks concerning the communion of saints and the association of ideas,

“The association is the idea.”
 

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

Friday, October 4, 2002

Friday October 4, 2002

Filed under: General — Tags: — m759 @ 4:17 AM

ART WARS:
The Agony and the Ya-Ya

Today's birthdays:

  • Charlton Heston
  • Anne Rice
  • Patti LaBelle

To honor the birth of these three noted spiritual leaders, I make the following suggestion: Use the mandorla as the New Orleans Mardi Gras symbol.  Rice lives in New Orleans and LaBelle's classic "Lady Marmalade" deals with life in that colorful city.

What, you may well ask, is the mandorla? This striking visual symbol was most recently displayed prominently at a meeting of U.S. cardinals in the Pope's private library on Shakespeare's birthday.  The symbol appears in the upper half of a painting above the Pope.

From Church Anatomy:

The illustration below shows how Barbara G. Walker in her excellent book "The Woman’s Encyclopedia of Myths and Secrets" describes the mandorla.

 

 

 

The Agony
and the Ecstasy

Based on a novel by Irving Stone, this 1965 movie focuses on the relationship between Michelangelo (Charlton Heston) and Pope Julius II (Rex Harrison), who commissioned the artist to paint the Sistine Chapel ceiling.

Vesica piscis

Mandorla, "almond," the pointed-oval sign of the yoni, is used in oriental art to signify the divine female genital; also called vesica piscis, the Vessel of the Fish. Almonds were holy symbols because of their female, yonic connotations.

Christian art similarly used the mandorla as a frame for figures of God, Jesus, and saints, because the artists forgot what it formerly meant. I. Frazer, G.B., 403

 

 

 
For further details on the mandorla (also known as the "ya-ya") see my June 12, 2002, note The Ya-Ya Monologues.
 
A somewhat less lurid use of the mandorla in religious art — the emblem of the Episcopal Diocese of South Carolina, taken from the website of St. Michael's Church in Charleston — is shown below.
 

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