This figure from a post of April 29, 2008 is related to remarks
by art theorist Rosalind Krauss in tonight's previous post.
Related language —
"Kernel" in mathematics and "Innerste Kern." elsewhere.
Related philosophy —
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
— Wittgenstein, Philosophical Investigations (1953), Section 109
The New York Times yesterday reported that Marxist theorist
Fredric Jameson died on Sunday.
Related material from a search for Jameson in this journal —
|
Rosalind Krauss in The Optical Unconscious
|
Today's host for a special political edition of CBS Sunday Morning
is Ted Koppel. Vocabulary review:
Koppel's appearance today was backed by the usual CBS Sunday Morning
sun-disk Apollo symbol. An Apollo symbol that some may prefer —
The Ninefold Square
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
The title is from . . .
https://booksonthewall.com/blog/samuel-beckett-quote-fail-better/ .
This post was suggested by yesterday's Feast of St. Thomas Becket —
A Beckett-related flashback linked to here yesterday —
"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity."
"If the window is this matrix of ambi- or multivalence,
and the bars of the windows-the grid-are what help us
to see, to focus on, this matrix, they are themselves
the symbol of the symbolist work of art. They function as
the multilevel representation through which the work of art
can allude, and even reconstitute, the forms of Being."
— Page 59, Rosalind Krauss, "Grids," MIT Press,
October , Vol. 9 (Summer, 1979), pp. 50-64
Related material —
Click the above image for a related Log24 post of 15 years ago today.
A related literary remark —
"Imagine Raiders of the Lost Ark set in 20th-century London, and then
imagine it written by a man steeped not in Hollywood movies but in Dante
and the things of the spirit, and you might begin to get a picture…."
— Doug Thorpe in an Amazon.com book review, not of Dark Materials.
Grids
Author: Rosalind Krauss
Source: October , Vol. 9 (Summer, 1979), pp. 50-64
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/778321
From page 59:
"Flowing and freezing; glace in French means glass, mirror, and ice;
transparency, opacity, and water. In the associative system
of symbolist thought this liquidity points in two directions.
First, towards the flow of birth-the amniotic fluid, the 'source'-
but then, towards the freezing into stasis or death-
the unfecund immobility of the mirror. For Mallarmé, particularly,
the window functioned as this complex, polysemic sign by which
he could also project the 'crystallization of reality into art.' 5
Mallarmé's Les Fenêtres dates from 1863;
Redon's most evocative window, Le Jour , appeared in 1891
in the volume Songes . If the window is this matrix of
ambi- or multivalence, and the bars of the windows-the grid-
are what help us to see, to focus on, this matrix, they are
themselves the symbol of the symbolist work of art.
They function as the multilevel representation through which
the work of art can allude, and even reconstitute, the forms of Being."
5 Robert G. Cohn, "Mallarmé's Windows," Yale French Studies ,
no. 54 (1977), 23-31.
Another evocative example — See Galois Window in this journal.
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
The Theory and
Organization of the
Bauhaus (1923)
A relevant illustration:
At math.stackexchange.com on March 1-12, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
These references will not appeal to those who enjoy modernism as a religion.
(For such a view, see Rosalind Krauss on grids and another writer's remarks
on the religion's 100th anniversary this year.)
Some related nihilist philosophy from Cormac McCarthy —
"Forms turning in a nameless void."
"… thinking starts with a problem, a difficulty, a contradiction."
— The late Arthur Mattuck of MIT in last night's post Gestalt .
"The 'technical support' is an underlying ground for aesthetic practice
that supports the work of art as canvas supported oil paint."
— MIT Press on the book by Rosalind Krauss titled Under Blue Cup .
From Grid View and List View (Log24, 15 April 2021) —
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
.jpg)
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
From AntiChristmas 2010 —
Above: Art Theorist Rosalind Krauss and The Ninefold Square
For Krauss:
Grid View List View
“Art bears the same relationship to society
that the dream bears to mental life. . . .
Like art, the dream mediates between order
and chaos. So, it is half chaos. That is why
it is not comprehensible. It is a vision, not
a fully fledged articulated production.
Those who actualize those half-born visions
into artistic productions are those who begin
to transform what we do not understand into
what we can at least start to see.”
— A book published on March 2, 2021:
Beyond Order , by Jordan Peterson
The inarticulate, in this case, is Rosalind Krauss:
A “raid on the inarticulate” published in Notices of the
American Mathematical Society in the February 1979 issue —
From this journal on Nov. 9-12, 2004:
|
Fade to Black “…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent.” — Trevanian, Shibumi “‘Haven’t there been splendidly elegant colors in Japan since ancient times?’ ‘Even black has various subtle shades,’ Sosuke nodded.” — Yasunari Kawabata, The Old Capital  Ad Reinhardt An Ad Reinhardt painting described in the entry of
Ad Reinhardt, Abstract Painting, 1960-66. The viewer may need to tilt the screen to see that “The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.  Greek Cross There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.” — “Grids,” by Rosalind Krauss, |
Related material from The New York Times today —
Remarks by Rosalind Krauss in the previous post suggest a look at …
Then there is the universal beauty of oneself :
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self.
See also "True Grid " in this journal.
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
See as well . . .
Prominent in the oeuvre of art theorist Rosalind Krauss, the Klein group
is a four-element group named for Felix Christian Klein.
It is commonly known as the four-group.
Mathematicians sometimes call this group
"V," for its German name, Vierergruppe .
For those who prefer narrative to mathematics —
(Namely, Plato's ghost)
Background: "Transcendental subject" is Kant's term for, more or less, the self.
"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity."
More from Krauss —
A book by an author with somewhat wider "cultural experience" —
See also "Plato's Diamond" in this journal.
Three hidden keys open three secret gates
Wherein the errant will be tested for worthy traits
And those with the skill to survive these straits
Will reach The End where the prize awaits
— Ready Player One , by Ernest Cline
Related text —
Tu es Petrus et super hanc petram
aedificabo ecclesiam meam et tibi
dabo claves regni caelorum
Related imagery —
From Steven Spielberg's film "Ready Player One" (2018) —
From this journal on June 17, 2003 —
From The New York Times on Easter night, 2007 —
See as well Rosalind Krauss on LeWitt:
An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —
Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:
"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity."
"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."
Illustration from a post of Feb. 3, 2011 —
.
The New York TImes reports this evening that
"Jon Underwood, Founder of Death Cafe Movement,"
died suddenly at 44 on June 27.
This journal on that date linked to a post titled "The Mystic Hexastigm."
A related remark on the complete 6-point from Sunday, April 28, 2013 —
(See, in Veblen and Young's 1910 Vol. I, exercise 11,
page 53: "A plane section of a 6-point in space can
be considered as 3 triangles perspective in pairs
from 3 collinear points with corresponding sides
meeting in 3 collinear points." This is the large
Desargues configuration. See Classical Geometry
in Light of Galois Geometry.)
This post was suggested, in part, by the philosophical ruminations
of Rosalind Krauss in her 2011 book Under Blue Cup . See
Sunday's post Perspective and Its Transections . (Any resemblance
to Freud's title Civilization and Its Discontents is purely coincidental.)
The title phrase is from Rosalind Krauss (Under Blue Cup , 2011) —
Another way of looking at the title phrase —
"A very important configuration is obtained by
taking the plane section of a complete space five-point."
(Veblen and Young, 1910, p. 39) —
For some context, see Desargues + Galois in this journal.
From a post of last Friday, June 2 —
See also Transformers in this journal.
"Before time began, there was the Cube."
— Transformers (2007)
According to art historian Rosalind Krauss in 1979,
the grid's earliest employers
"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."
The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:
See instances of the above image.
The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —
The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.
Two news items on art as a tool:
Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:
Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:
Which is the tool here, the grid or Krauss?
Barron's Educational Series (click to enlarge):
The Tablet of Ahkmenrah:
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
Another educational series (this journal):
Art theorist Rosalind Krauss and The Ninefold Square
Once again, Harvard defeats Holy Cross.
See also a related remark by Norman Mailer, and Plan 9 in this journal.
Presumably the Holy Cross defeat will please art theorist Rosalind Krauss (below).
(Click to enlarge.)
A prequel to today's noon post
Rosalind Krauss in "Grids" —
"The physical qualities of the surface, we could say,
are mapped onto the aesthetic dimensions of
the same surface. And those two planes—
the physical and the aesthetic— are demonstrated
to be the same plane…. the bottom line of the grid is
a naked and determined materialism.”
A writer I prefer:
Barsotti's classic illustrated:
The naked and determined Annette Bening in "The Grifters" —
“. . . the bottom line of the grid is a naked and determined materialism.”
— Rosalind Krauss, quoted by Josefine Lyche
See also http://www.dailymotion.com/video/
x164rmi_britt-ekland-nude-wicker-man-1973_people.
“Chaos is order yet undeciphered.”
— The novel The Double , by José Saramago,
on which the recent film "Enemy" was based
For Louise Bourgeois — a post from the date of Galois's death—
For Toronto — Scene from a film that premiered there on Sept. 8, 2013:
Related material: This journal on that date, Sept. 8, 2013:
"I still haven't found what I'm looking for." — Bono
"In fact Surrealism found what it had been looking for
from the first in the 1920 collages [by Max Ernst],
which introduced an entirely original scheme of
visual structure…."
— Rosalind Krauss quoting André Breton*
in "The Master's Bedroom"
* "Artistic Genesis and Perspective of Surrealism"
(1941), in Surrealism and Painting (New York,
Harper & Row, 1972, p. 64).
See also Damnation Morning in this journal.
For John Cusack and Menno Meyjes —
"I still haven't found what I'm looking for." — Bono
"In fact Surrealism found what it had been looking for
from the first in the 1920 collages [by Max Ernst],
which introduced an entirely original scheme of visual structure…."
— Rosalind Krauss quoting André Breton*
in "The Master's Bedroom"
See also tonight's 10 PM post.
* "Artistic Genesis and Perspective of Surrealism" (1941),
in Surrealism and Painting (New York, Harper & Row, 1972, p. 64).
… For the Harvard Arts Weekend:
"Grids, You Say?" by Josefine Lyche, with
Lyche's quotation from Rosalind Krauss in October
(Vol. 9, Summer 1979) —
 |
See also last evening's Elevation of the Host, with Vampire Weekend.
"For every kind of vampire, there is a kind of cross." — Gravity's Rainbow
From AntiChristmas 2010—
Art theorist Rosalind Krauss and The Ninefold Square
Krauss is a co-founder of the art journal October .
For some backgound, see the overlapping searches
Krauss October and Matrix Architect.
"In the master's bedroom, they gathered for the feast…."
— Suggested by the current film Hotel Transylvania
"For every kind of vampire, there is a kind of cross."
– Thomas Pynchon, Gravity's Rainbow
Related material— the Feast of Saint Patrick in 2009.
For Trotsky's Birthday (Old Style), 2009—
Related material:
(Click for further details.)
See also St. Stephen's Day, 2011.
(The title is a nod to Peter Woit's recent post "Nothingness Smackdown.")
"To wrestle new mediums to the mat of specificity has been a preoccupation of mine since the inception of October , the magazine I founded in 1976 with Annette Michelson, the first issue of which carried my essay 'Video and Narcissism' which attempts to tie the essence of video to the spectacular nature of mirrors."
— Rosalind Krauss, 2008, introduction to Perpetual Inventory (MIT Press, 2010)
Related material— The video art and mirror art of Josefine Lyche.
See also Krauss's essay on video in Perpetual Inventory— "Video: The Aesthetics of Narcissism" (first published as "Video and Narcissism," October , no. 1 (Spring 1976))—
"In The Language of the Self , Lacan begins by characterizing the space of the therapeutic transaction as an extraordinary void created by the silence of the analyst. Into this void the patient projects the monologue of his own recitation, which Lacan calls 'the monumental construct of his narcissism.'"
— and related remarks on October and the void quoted here March 10 in "Boo Boo Boo."
"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."
— Rosalind Krauss in "Grids,"
October (Summer 1979), 9: 50-64.
Counterexample—
The Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
| Mathematics and Narrative, Illustrated |
|
 Mathematics |
 Narrative |
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the Dolgachev-Artebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
|
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
|
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
|
Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, |
And from last October —
|
Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— |
The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."
That search was suggested by the title of today's New York Times op-ed essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.
Further information on one of the images above—
A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics yields the following—
|
In defense of Plato’s realism (vs. sophists’ nominalism– see recent entries.) Plato cited geometry, notably in the Meno , in defense of his realism.
|
For the Meno 's diamond figure in Giaquinto, see a review—
— Review by Jeremy Avigad (preprint)
Finite geometry supplies a rather different context for Plato's "basic picture."
In that context, the Klein four-group often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)
The Times op-ed essay today notes that linguistic translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."
The list of four-group transformations in the mathematical sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.
Between midnight and dawn, when the past is all deception,
The future futureless, before the morning watch
When time stops and time is never ending….
Art Theorist Rosalind Krauss and The Ninefold Square
* Title from a quotation in the Sunday New York Times today
(in a story about a June 25 death beginning on page MB1 of the New York edition)
John Allen Paulos yesterday at Twitter—
"Plato's code cracked? http://bit.ly/ad6k1S
Fascinating if not a hoax or hype."
The story that Paulos linked to is about a British
academic who claims to have found some
symbolism hidden in Plato's writings by
splitting each into 12 parts and correlating
the 12 parts with semitones of a musical scale.
I prefer a different approach to Plato that is
related to the following hoax and hype—
HOAX:
From Dan Brown's novel Angels & Demons (2000)—
HYPE:
This four-elements diamond summarizes the classical
four elements and four qualities neatly, but some scholars
might call the figure "hype" since it deals with an academically
disreputable subject, alchemy, and since its origin is unclear.
For the four elements' role in some literature more respectable
than Dan Brown's, see Poetry's Bones.
Although an author like Brown might spin the remarks
below into a narrative— The Plato Code — they are
neither hoax nor hype.
NOT HOAX:
NOT HYPE:
For related non-hoax, non-hype remarks, see
The Rational Enterprise: Logos in Plato's Theaetetus,
by Rosemary Desjardins.
Those who prefer hoax and hype in their philosophy may consult
the writings of, say, Barbara Johnson, Rosalind Krauss, and—
in yesterday's NY Times's "The Stone" column— Nancy Bauer.
— The New York Times
Art Theorist Rosalind Krauss and The Ninefold Square
Friday’s post “Religion at Harvard” continues…
This list may be of some use to
Rebecca Newberger Goldstein, who, like Prothero,
spoke recently at Harvard Book Store.
See also Rosalind Krauss on Grids,
An Education, and Plan 9 from Outer Space.
Readers more advanced than Harvard audiences
may wish to compare yesterday’s linked-to story
“Loo Ree” with the works of Alison Lurie—
in particular, Imaginary Friends and Familiar Spirits.
|
January 06, 2007 Picture of Nothing
“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….” Related material: The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.” — Tom Wolfe, The Painted Word “Concept (scholastics’ verbum mentis)– theological analogy of Son’s procession as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957, second printing 1963, page 162
“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”
— Jim Holt, Big-Bang Theology, from Slate‘s “High Concept” department  “Bang.” “…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….” For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.
|
Some Context:
See also Nativity.
"We have named this journal in celebration of that moment in our century when revolutionary practice, theoretical inquiry and artistic innovation were joined in a manner exemplary and unique." — Rosalind Krauss and Annette Michelson, "About October," October (Spring, 1976, MIT Press) 1: pp. 3–5
Yesterday’s entry Deep Structures discussed the “semiotic square,” a device that exemplifies the saying “If you can’t dazzle ’em with brilliance, then baffle ’em with bullshit.”
A search today for what the Marxist critic Fredric Jameson might have meant by saying that the square “is capable of generating at least ten conceivable positions out of a rudimentary binary opposition” leads to two documents of interest.
1. “Theory Pictures as Trails: Diagrams and the Navigation of Theoretical Narratives” (pdf), by J.R. Osborn, Department of Communication, University of California, San Diego (Cognitive Science Online, Vol.3.2, pp.15-44, 2005)
2. “The Semiotic Square” (html), by Louis Hébert (2006), professor, Université du Québec à Rimouski, in Signo (http://www.signosemio.com).
Shown below is Osborn’s picture of the semiotic square:
On the brighter side, we have, as a sign that Gallic clarity still exists, the work of Hébert.
Here is how he approaches Jameson’s oft-quoted, but seemingly confused, remark about “ten conceivable positions”–
“The Semiotic Square,”
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5. (=1+2) COMPLEX TERM
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7. (=1+3) POSITIVE DEIXIS |
8. (=2+4) NEGATIVE DEIXIS
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6. (=3+4) NEUTRAL TERM
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LEGEND:
The + sign links the terms that are combined to make up a metaterm (a compound term); for example, 5 is the result of combining 1 and 2.
The semiotic square entails primarily the following elements (we are steering clear of the constituent relationships of the square: contrariety, contradiction, and complementarity or implication):
1. terms
2. metaterms (compound terms)
3. object(s) (classified on the square)
4. observing subject(s) (who do the classifying)
5. time (of the observation)
The semiotic square is composed of four terms:
Position 1 (term A)
Position 2 (term B)
Position 3 (term not-B)
Position 4 (term not-A)
The first two terms form the opposition (the contrary relationship) that is the basis of the square, and the other two are obtained by negating each term of the opposition.
The semiotic square includes six metaterms. The metaterms are terms created from the four simple terms. Some of the metaterms have been named. (The complex term and the neutral term, despite their names, are indeed metaterms).
Position 5 (term 1 + term 2): complex term
Position 6 (term 3 + term 4): neutral term
Position 7 (term 1 + term 3): positive deixis
Position 8 (term 2 + term 4): negative deixis
Position 9 = term 1 + term 4: unnamed
Position 10 = term 2 + term 3: unnamed
These ten “positions” are apparently meant to explain Jameson’s remark.
Hébert’s treatment has considerably greater entertainment value than Osborn’s. Besides “the living dead” and angels, Hébert’s examples and exercises include vampires, transvestites, the Passion of Christ, and the following very relevant quotation:
“Simply let your ‘Yes’ be ‘Yes,’ and your ‘No,’ ‘No’; anything beyond this comes from the evil one.” (Matthew 5:37)
The Square of Oppositon
at Stanford Encylopedia of Philosophy
The Square of Opposition
in its original form
"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."
— Edward Buckner at The Logic Museum
From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:
The Semiotic Square
"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."
* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter
Another version of the semiotic square:
Here is a more explicit figure representing the Klein group:
There is also the logical
diamond of opposition —
A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
a Marxist literary theorist:
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"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–
The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.) Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s1) for example is understood in relation to its contrary, "death" (s2). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (-S), i.e., "not-life" (-s1) and "not-death" (-s2). We would therefore be left with the following semiotic square (Fig. 1): 
As Jameson explains in the Foreword to Greimas' On Meaning, "-s1 and -s2"—which in this example are taken up by "not-death" and "not-life"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, not-life would include more than merely death and not-death more than life.
* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976.
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— The Gameplayers of Zan, by M.A. Foster
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
Gravity's Rainbow
Crosses used by semioticians
to baffle their opponents
are illustrated above.
Some other kinds of crosses,
and another kind of opponent:
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Monday, July 11, 2005
Logos
for St. Benedict's Day Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.
Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth. Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion. I personally prefer the "diamond theory" of truth, represented by the logo below.
See also the previous entry Sunday, July 10, 2005
Mathematics
and Narrative Click on the title for a narrative about
Nikolaos K. Artemiadis,
"First of all, I'd like to
— Remark attributed to Plato
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||||||||
mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.
In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago —
There is
another sort of object
we may associate with a
different museum and with
a modern Constantine …
See "Art Wars for MoMA"
(Dec. 14, 2008).
This object, modern
rather than medieval,
is the ninefold square:
It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.
For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:
Selah.
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The following meditation was inspired by the recent fictional recovery, by Mira Sorvino in "The Last Templar," of a Greek Cross — "the Cross of Constantine"– and by the discovery, by art historian Rosalind Krauss, of a Greek Cross in the art of Ad Reinhardt. |
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Note that in applications, the vertical axis of the Cross of Descartes often symbolizes the timeless (money, temperature, etc.) while the horizontal axis often symbolizes time. T.S. Eliot:
"Men’s curiosity searches past and future |
|
"In suggesting that the success [1] of the grid
[1] Success here refers to
three things at once: a sheerly quantitative success, involving the number of artists in this century who have used grids; a qualitative success through which the grid has become the medium for some of the greatest works of modernism; and an ideological success, in that the grid is able– in a work of whatever quality– to emblematize the Modern." — Rosalind Krauss, "Grids" (1979) |
Related material:
Time Fold and Weyl on
objectivity and frames of reference.
See also Stambaugh on
The Formless Self
as well as
A Study in Art Education
and
Jung and the Imago Dei.
A Sunrise
for Sunrise
“If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.”
Yesterday’s entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo)
may be used to create a graphic design– a Greek cross, which appears also in crossword puzzles:
Illustration by
Paul Rand
(born Peretz Rosenbaum)
A more sophisticated example
of the ninefold square
in graphic design:
“That old Jew
gave me this here.”
— A Flag for Sunrise
Raiders of
the Lost Well
| “The challenge is to keep high standards of scholarship while maintaining showmanship as well.” |
— Olga Raggio, a graduate of the Vatican library school and the University of Rome who, at one point in her almost 60 years with the Metropolitan Museum of Art, organized “The Vatican Collections,” a blockbuster show. Dr. Raggio died on January 24.
The next day, “The Last Templar,” starring Mira Sorvino, debuted on NBC.
“One highlight of Secretary of State Hillary Clinton’s first overseas trip will be a stop in China. Her main mission in Beijing will be to ensure that US-China relations under the new Obama administration get off to a positive start.”
— Stephanie Ho, Voice of America Beijing bureau chief, today
Symbol of The Positive,
from this journal
on Valentine’s Day:
Stephanie was born in Ohio and grew up in California. She has a bachelor’s degree in Asian studies with an emphasis on Chinese history and economics, from the University of California at Berkeley.”
“She is fluent in
Mandrin Chinese.”
—VOA
As is Mira Sorvino.
Those who, like Clinton, Raggio, and
Sorvino’s fictional archaeologist in
“The Last Templar,” prefer Judeo-
Christian myths to Asian myths,
may convert the above Chinese
“well” symbol to a cross
(or a thick “+” sign)
by filling in five of
the nine spaces outlined
by the well symbol.
In so doing, they of course
run the risk, so dramatically
portrayed by Angelina Jolie
as Lara Croft, of opening
Pandora’s Box.
(See Rosalind Krauss, Professor
of Art and Theory at Columbia,
for scholarly details.)
Krauss
— John Murphy at Bardolatry.com on November 21, 2008
A vast library…
On searching for Garden of Eden patterns (GEP's):
"The grid is a staircase to the Universal…."
— Rosalind Krauss, quoted here on Weyl's birthday, 2004
"I find the whole topic of GEPs a deeply interesting one, from many viewpoints: mathematical, philosophical, physical….
… the obvious problem is, that the required computational time is growing rapidly with the size of the grid, and even for a small grid, like 4×4 (=16 cells) there are 216=65536 possible patterns…."
— cateye at RichardDawkins.net
… and magical powers
The date of cateye's post was Sunday, October 21, 2007.
For related material see Log24 on Sunday, October 21, 2007.
Thomas Wolfe
(Harvard M.A., 1922)
versus
Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)
on
"No culture has a pact with eternity."
— George Steiner, interview in
The Guardian of
"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."
— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005
From today's online Harvard Crimson:
"… under the leadership of Faust,
Harvard students should look forward
to an ever-growing opportunity for
international experience
and artistic endeavor."
Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen
From a recent book
on Wolfgang Pauli,
The Innermost Kernel:
A belated happy birthday
to the late
Felix Christian Klein
(born on April 25) —
Another Harvard figure quoted here on Dec. 5, 2002:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)
From a review of Rosalind Krauss's The Optical Unconscious (MIT Press hardcover, 1993):
Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid-19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."
— Arthur C. Danto in ArtForum, Summer 1993
Rosalind Krauss in The Optical Unconscious (MIT Press paperback, 1994):
For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude Lévi-Strauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.
Wikipedia on the Klein group (denoted V, for Vierergruppe):
In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.
For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:
Kernel
Mathematical Reviews citation:
MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9
A quote from MR at Amazon.com:
"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
The four-group is also known as the Vierergruppe or Klein group. It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)
The "chessboard" of
Nov. 7, 2006
(as revised Nov. 7, 2012)–
(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
"… Mondrian and Malevich
are not discussing canvas
or pigment or graphite
or any other form of matter.
They are talking about about
Being or Mind or Spirit.
From their point of view,
the grid is a staircase
to the Universal…."
(continued from
January 9, 2003)
George Balanchine
|
"What on earth is
a concrete universal?"
— Robert M. Pirsig
Review:
From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:
"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."
There is, however,
"the field of reason"–
the 3×3 grid:
Click on grid
for details.
As Rosalind Krauss
has noted, some artists
regard the grid as
"a staircase to
the Universal."
Other artists regard
Epiphany itself as an
approach to
the Universal:
— Richard Kearney, 2005,
in The New Arcadia Review
Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.
— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 2005-2006
For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:
|
"We symbolize
logical necessity with the box  )and logical possibility with the diamond  )."
"The possibilia that exist,
— Michael Sudduth, |
Picture of Nothing
On Kirk Varnedoe’s
2003 Mellon Lectures,
“Pictures of Nothing“–
“Varnedoe’s lectures were ultimately
about faith, about his faith in
the power of abstraction,
and abstraction as a kind of
anti-religious faith in itself….”
Related material:
The more industrious scholars
will derive considerable pleasure
from describing how the art-history
professors and journalists of the period
1945-75, along with so many students,
intellectuals, and art tourists of every
sort, actually struggled to see the
paintings directly, in the old
pre-World War II way,
like Plato’s cave dwellers
watching the shadows, without
knowing what had projected them,
which was the Word.”
— Tom Wolfe, The Painted Word
“Concept (scholastics’ verbum mentis)–
theological analogy of Son’s procession
as Verbum Patris, 111-12″
— Index to Joyce and Aquinas,
by William T. Noon, S.J.,
Yale University Press 1957,
second printing 1963, page 162
“So did God cause the big bang?
Overcome by metaphysical lassitude,
I finally reach over to my bookshelf
for The Devil’s Bible.
Turning to Genesis I read:
‘In the beginning
there was nothing.
And God said,
‘Let there be light!’
And there was still nothing,
but now you could see it.'”
— Jim Holt, Big-Bang Theology,
Slate‘s “High Concept” department
“Bang.”
“…Mondrian and Malevich
are not discussing canvas
or pigment or graphite or
any other form of matter.
They are talking about
Being or Mind or Spirit.
From their point of view,
the grid is a staircase
to the Universal….”
For properties of the
“nothing” represented
by the 3×3 grid, see
The Field of Reason.
For religious material related
to the above and to Epiphany,
a holy day observed by some,
see Plato, Pegasus, and the
Evening Star and Shining Forth.
Rosalind Krauss
in "Grids," 1979:
"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.
Or, to take a more up-to-date example…."
"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
— The Club Dumas,1993
"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
— Robert Plant,1971
The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.
An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:
"This is the garden of Apollo,
the field of Reason…."
— John Outram, architect
For more on the field
of reason, see
Log24, Oct. 9, 2006.
A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).
Unreason is, of course,
more popular.
* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:
"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon
Pandora's Box
Part I:
The Pandora Cross
— Rosalind Krauss in "Grids"
Part II:
The Opening
Remarks by the Pope on Sept. 12,
as reported by the Vatican:
Faith, Reason, and the University:
Memories and Reflections
For the result of
the Pope's remarks, see
a transcript of
yesterday's Google News
and the following
from BBC today:
Click to enlarge the screenshot.
Part III:
Hope
"In September [2005], she had a private audience with Pope Benedict XVI at Castel Gandolfo, his summer residence outside Rome. She had criticized John Paul II for making overtures to Muslims, and for not condemning terrorism heartily enough, but she has hopes for Joseph Ratzinger."
For further details, see yesterday's Log24.
Part IV:
The Sibyl's Song
— From The Magic Circle,
a spiritual narrative
by Katherine Neville
For more on "the long-mute voice
of the past," on "darkness beneath
the volcano," and on uncorking,
see Glory Season and Harrowing.
Related material from
Log24 on Dec. 2, 2005:
Benedict XVI, before he became Pope:
and a related
Christian symbol,
the Greek Cross
(adapted from
Ad Reinhardt).
Moral of the Pandora Cross:
"Nine is a very powerful Nordic number."
— Katherine Neville in The Magic Circle…
quoted in The Nine, a Log24 entry
for Hermann Weyl's birthday,
November 9, 2004.
The Krauss Cross
"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.
Or, to take a more up-to-date example, we could think about Ad Reinhardt who, despite his repeated insistence that 'Art is art,' ended up by painting a series of black nine-square grids in which the motif that inescapably emerges is a Greek cross. There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."
Rebecca Goldstein on
Mathematics and Narrative:
"I don't write exclusively on Jewish themes or about Jewish characters. My collection of short stories, Strange Attractors, contained nine pieces, five of which were, to some degree, Jewish, and this ratio has provided me with a precise mathematical answer (for me, still the best kind of answer) to the question of whether I am a Jewish writer. I am five-ninths a Jewish writer."
Jacques Maritain,
October 1941:
"The passion of Israel
today is taking on
more and more distinctly
the form of the Cross."
E. L. Doctorow,
City of God:
"In the garden of Adding,
Live Even and Odd."
ART WARS
continued from
previous entry
In memory of
Elisabeth Schwarzkopf:
This question might
well be posed by…
Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University
(Ph.D., Harvard U., 1969).
"The grid is a staircase to the Universal….
We could think about Ad Reinhardt, who,
despite his repeated insistence that
'Art is art,'
ended up by painting a series of…
nine-square grids in which the motif
that inescapably emerges is
a Greek cross.
Adapted from
Ad Reinhardt
There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape and the
Pandora's box of spiritual reference
that is opened once one uses it."
— Rosalind Krauss in "Grids"
"Nine is a very powerful Nordic number."
— Katherine Neville, author of The Eight
Related material:
Born today: Arnold Schwarzenegger
Frame not included in
Terminator 2: Judgment Day
"…Mondrian and Malevich are not
discussing canvas or pigment or graphite
or any other form of matter.
They are talking about
Being or Mind or Spirit.
From their point of view, the grid
is a staircase to the Universal…."
— Rosalind Krauss, "Grids"
Born today: Hilary Swank
"The grid is a staircase to the Universal."
— Rosalind Krauss
"To live is to defend a form."
— attributed to Hölderlin
|
Fade to Black "…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent." — Trevanian, Shibumi "'Haven't there been splendidly elegant colors in Japan since ancient times?' 'Even black has various subtle shades,' Sosuke nodded." — Yasunari Kawabata, The Old Capital
An Ad Reinhardt painting
Ad Reinhardt,
The viewer may need to tilt "The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that 'Art is art,' ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.
Greek Cross There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."
— Rosalind Krauss,
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In memory of
St. William Golding
(Sept. 19, 1911 – June 19, 1993)
“Horton Marlais Davies, Putnam professor emeritus of religion at Princeton and an author of many books about church history, died on Wednesday at his home in Princeton, N.J. He was 89…. Dr. Davies specialized in the impact of Christianity on the arts.”
The Catholic Encyclopedia (1908) on the communion of saints:
“One cannot read the parables of the kingdom (Matt., xiii) without perceiving its corporate nature and the continuity which links together the kingdom in our midst and the kingdom to come. The nature of that communion, called by St. John a fellowship with one another (‘a fellowship with
Related material:
Religious art in the entry Art History of 11 AM Wednesday, May 11, the date of Davies’s death. See also the following direct and indirect links from that entry:
To a cruciform artifact from the current film Kingdom of Heaven, to an entry quoting John xv, Nine is a Vine, and to Art Theory for Yom Kippur.
For less-religious material on the number nine, see the entries and links in the Log24 archive for June 17-30, 2004.
“If we open any tract–
Plastic Art and Pure Plastic Art
or The Non-Objective World,
for instance– we will find that
Mondrian and Malevich are not
discussing canvas or pigment or
graphite or any other form of
matter. They are talking about
Being or Mind or Spirit.”
Amen.
Reuters – "Joe Grant, a legendary Disney artist who designed the Queen/Witch in 'Snow White and the Seven Dwarfs,' died of a heart attack while doing what he loved most, drawing, the Walt Disney Co. said Monday.
Grant, 96, died at his home in the Los Angeles suburb of Glendale last Friday while sitting at his drawing board."
— Opening sentence of
Martha Cooley's The Archivist
From Log24 last Friday,
a Greek cross:
Click on picture for details.
And from Sunday, May 1
(Orthodox Easter):
Columbia University's
Meyer Schapiro Professor
of Modern Art and Theory:
"There is no painter in the West
who can be unaware of
the symbolic power of
the cruciform shape1
and the Pandora's box
of spiritual reference2
that is opened
once one uses it."
1, 2, 3 Today's birthdays:
1 Natasha Richardson, born 11 May 1963,
Jedi wife and costar of Nell
2 Martha Quinn, born 11 May 1959,
MTV wit
3 Frances Fisher, born 11 May 1952,
dazzling redhead
"Professor Krauss even uses many of the same decorations with which she festooned earlier volumes. Bataille’s photograph of a big toe, for example, which I like to think of as her mascot, reappears. As does her favorite doodle, a little graph known as a 'Klein Group' or 'L Schema' whose sides and diagonals sport arrows pointing to corners labeled with various opposing pairs: e.g., 'ground' and 'not ground,' 'figure' and 'not figure.' Professor Krauss seems to believe that this device, lifted from the pages of structuralist theory, illuminates any number of deep mysteries: the nature of modernism, to begin with, but also the essence of gender relations, self-consciousness, perception, vision, castration anxiety, and other pressing conundrums that, as it happens, she has trouble distinguishing from the nature of modernism. Altogether, the doodle is a handy thing to have around. One is not surprised that Professor Krauss reproduces it many times in her new book."
 A Jungian on this six-line figure: "They are the same six lines |
"People have believed in the fundamental character of binary oppositions since at least classical times. For instance, in his Metaphysics Aristotle advanced as primary oppositions: form/matter, natural/unnatural, active/passive, whole/part, unity/variety, before/after and being/not-being.* But it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle's Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality….
The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully…."
— Daniel Chandler, Semiotics for Beginners.
* Compare Chandler's list of Aristotle's primary oppositions with Aristotle's list (also in the Metaphysics) of Pythagorean oppositions (see Midrash Jazz Quartet).
Harvard's Barry Mazur on
one mathematical style:
"It’s the barest, most Beckett-like vocabulary
that incorporates the theory and nothing else."
Samuel Beckett, Quad (1981):
A Jungian on this six-line logo:
"They are the same six lines
that exist in the I Ching….
Now observe the square more closely:
four of the lines are of equal length,
the other two are longer….
For this reason symmetry
cannot be statically produced
and a dance results."
— Marie-Louise von Franz,
Number and Time (1970),
Northwestern U. Press
paperback, 1979, p. 108
A related logo from
Columbia University's
Department of Art History
and Archaeology:
Also from that department:
Meyer Schapiro Professor
of Modern Art and Theory:
"There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape
and the Pandora's box
of spiritual reference
that is opened
once one uses it."
"In the garden of Adding
live Even and Odd…"
— The Midrash Jazz Quartet in
City of God, by E. L. Doctorow
THE GREEK CROSS
Here, for reference, is a Greek cross
within a nine-square grid:
Related religious meditation for
Doctorow's "Garden of Adding"…
Types of Greek cross
illustrated in Wikipedia
under "cross":
THE BAPTISMAL CROSS
Related material:
Fritz Leiber's "spider"
or "double cross" logo.
See Why Me? and
A Shot at Redemption.
Happy Orthodox Easter.
The Nine
(Readings for
Weyl’s birthday)
“The grid is a staircase to the Universal….
We could think about Ad Reinhardt, who,
despite his repeated insistence that
‘Art is art,’
ended up by painting a series of…
nine-square grids in which the motif
that inescapably emerges is
a Greek cross.
Greek Cross
There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape and the
Pandora’s box of spiritual reference
that is opened once one uses it.”
— Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University
(Ph.D., Harvard U., 1969),
in “Grids”
“Nine is a very powerful Nordic number.”
— Katherine Neville, author of The Eight,
in The Magic Circle,
Ballantine paperback,
1999, p. 339
“To live is to defend a form.”
(“Leben, das heisst eine Form verteidigen“)
— attributed to Hölderlin
For details on the above picture,
which deals with properties of the
nine-square grid, see
For more on the defense
of this form,
see the Log24.net entry of
June 5, 2004, A Form,
and the Art Wars entries
for St. Peter’s Day, 2004.
ART WARS:
Judgment Day
“…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….”
— Rosalind Krauss, “Grids”
Krauss is the Meyer Schapiro Professor of Modern Art and Theory at Columbia University.
For more on Meyer Schapiro, see the link on the phrase “art historian” in my March 10, 2003, entry.
To view that entry in a larger context, see the web page Art at the Vanishing Point, which includes a picture of Mondrian’s own Paris staircase. The picture below might be thought of as illustrating Krauss’s “grid is a staircase”… a staircase to, in fact, a vanishing point.
Frame not included in
Terminator 2: Judgment Day
For a different view of what the New York Times Book Review has characterized as “high culture,” see the link on that phrase also in my March 10, 2003, entry. This leads to a work by T. S. Eliot titled Christianity and Culture. See too the remarks of the Meyer Schapiro Professor in my Oct. 5, 2003, entry, “Art Theory for Yom Kippur,” in which she likens the Cross to Pandora’s box.
Eliot’s attitude toward this Jewish approach to high culture might be summarized by the following remarks of Sarah Connor in Terminator 2: Judgment Day—
Dr. Silberman: You broke my arm!
Sarah Connor: There are two-hundred-fifteen bones in the human body, [expletive deleted]. That’s one.
At Mount Sinai:
Art Theory for Yom Kippur
From the New York Times of Sunday, October 5, 2003 (the day that Yom Kippur begins at sunset):
“Rabbi Ephraim Oshry, whose interpretations of religious law helped sustain Lithuanian Jews during Nazi occupation…. died on Sept. 28 at Mount Sinai Hospital in Manhattan. He was 89.”
For a fictional portrait of Lithuanian Jews during Nazi occupation, see the E. L. Doctorow novel City of God.
For meditations on the spiritual in art, see the Rosalind Krauss essay “Grids.” As a memorial to Rabbi Oshry, here is a grid-based version of the Hebrew letter aleph:
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|
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Click on the aleph for details.
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God, by E. L. Doctorow
Here are two meditations
on Even and Odd for Yom Kippur:
Meditation I
From Rosalind Krauss, “Grids”:
“If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.
Or, to take a more up-to-date example, we could think about Ad Reinhardt who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of black nine-square grids in which the motif that inescapably emerges is a Greek cross. There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.”
Meditation II
Here, for reference, is a Greek cross
within a nine-square grid:
Related religious meditation for
Doctorow’s “Garden of Adding”…
ART WARS:
Time and the Grid
Art theorist Rosalind Krauss and poet T. S. Eliot on time, timelessness, and the grid.
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
“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.
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