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.

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.

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.

(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."

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….

— Four Quartets

June 25, 2010 —

Art Theorist Rosalind Krauss and The Ninefold Square

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:

Osborn’s discussion of the square, though more clear than, say, that of Rosalind Krauss (who reverses the bottom two parts of the square– see Deep Structures), fails. His Appendix A is miserably obscure.

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,”   by Louis Hébert 1. ABSTRACT The semiotic square, developed by Greimas and Rastier, is a means of refining oppositional analyses by increasing the number of analytical classes stemming from a given opposition from two (life/death, for instance) to four (for example, life, death, life and death (the living dead), and neither life nor death (angels)) to eight or even ten. 2. THEORY The actantial model, isotopy and the semiotic square are undoubtedly the best-known theoretical propositions that have emerged from the Paris School of semiotics, with Greimas as its central figure. Like the actantial model and the veridictory square, the semiotic square is designed to be both a conceptual network and a visual representation of this network, usually depicted in the form of a “square” (which actually looks like a rectangle!). Courtés defines it as the visual representation of the logical structure of an opposition (cf. Courtés, 1991, 152). The semiotic square is a means of refining oppositional analyses by increasing the number of analytical classes stemming from a given opposition from two (for instance, life/death) to four (for example, life, death, life and death (the living dead), and neither life nor death (angels)) to eight or even ten. Here is an empty semiotic square. Structure of the semiotic square     5. (=1+2) COMPLEX TERM       1. TERM A   2. TERM B   9. (=1+4) 10. (=2+3)   3. TERM NOT-B   4. TERM NOT-A   7. (=1+3) POSITIVE DEIXIS 8. (=2+4) NEGATIVE DEIXIS         6. (=3+4) NEUTRAL TERM     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. 2.1 CONSTITUENT ELEMENTS 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) 2.1.1 TERMS 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. 2.1.2 METATERMS 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)

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."

— Rosalind Krauss, "Grids"

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  

From Paul-Rand.com

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   
 

ART WARS continued:

The Krauss Cross

Rosalind Krauss in "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."

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:

"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:

Balanchine's Birthday,

Apollo and Christ.
 

Today’s New York Times:

“Horton Marlais Davies, Putnam professor emeritus of religion at Princeton and an author of many books about church history, died on Wednesday at his home in Princeton, N.J. He was 89…. Dr. Davies specialized in the impact of Christianity on the arts.”

The Catholic Encyclopedia (1908) on the communion of saints:

“One cannot read the parables of the kingdom (Matt., xiii) without perceiving its corporate nature and the continuity which links together the kingdom in our midst and the kingdom to come. The nature of that communion, called by St. John a fellowship with one another (‘a fellowship with us’ — I John, i, 3) because it is a fellowship with the Father, and with his Son, and compared by him to the organic and vital union of the vine and its branches (John, xv), stands out….”
 
Related material:

Religious art in the entry Art History of 11 AM Wednesday, May 11, the date of Davies’s death.  See also the following direct and indirect links from that entry:

To a cruciform artifact from the current film Kingdom of Heaven, to an entry quoting John xv, Nine is a Vine, and to Art Theory for Yom Kippur.

For less-religious material on the number nine, see the entries and links in the Log24 archive for June 17-30, 2004.

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):

Rosalind Krauss,

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."

"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."
 

From Drid Williams,
The Semiotics of Human Action,
Ritual, and Dance:

This is closely related to
Beckett's "Quad" figure

A Jungian on this six-line figure: "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)

and to the Greimas "semiotic square":

"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).
 

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”

Krauss

“Nine is a very powerful Nordic number.”
— Katherine Neville, author of The Eight,

in The Magic Circle,
Ballantine paperback,
1999, p. 339

Neville

“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

Translation Plane.

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.

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

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.