By Steven H. Cullinane and Websim.ai —
https://websim.ai/p/_dorlgddf3vcxh6qjysy/16 .
Saturday, October 26, 2024
Saturday, February 3, 2024
For Medium Man in February
"Who can pick up the weight of Britain,
Who can move the German load
Or say to the French here is France again?
Imago. Imago. Imago.
It is nothing, no great thing, nor man
Of ten brilliancies of battered gold
And fortunate stone. It moves its parade
Of motions in the mind and heart,
A gorgeous fortitude. Medium man
In February hears the imagination's hymns
And sees its images, its motions
And multitudes of motions…."
— From Wallace Stevens, "Imago."
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Saturday, June 19, 2010
Imago Creationis
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
Four-Part Tesseract Divisions—
The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to four-part partitions
of the 16 points in a finite Galois space
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Euclidean spaces versus Galois spaces in a larger context— Infinite versus Finite The central aim of Western religion —
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy —
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." |
Another picture related to philosophy and religion—
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
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Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61) |
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
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… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.” If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multi-dimensional symbols beyond our grasp. |
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
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Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. |
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Tuesday, September 11, 2007
Tuesday September 11, 2007
Battlefield Geometry
"The general, who wrote the Army's book on counterinsurgency, said he and his staff were 'trying to do the battlefield geometry right now' as he prepared his troop-level recommendations."
— Steven R. Hurst, The Associated Press, Wednesday, Aug. 15, 2007
"'… we are in the process of doing the battlefield geometry to determine the way ahead.'"
— Charles M. Sennott, Boston Globe, Friday, Sept. 7, 2007
"Based on these considerations, and having worked the battlefield
— United States Army, Monday, Sept. 10, 2007
Related material:
Log24 entries of
June 11 and 12, 2005:
"In the desert you can
remember your name
'Cause there ain't no one
for to give you no pain."
Saturday, June 4, 2005
Saturday June 4, 2005
Drama of the Diagonal
The 4×4 Square:
French Perspectives
The 4×4 Square:
French Perspectives
Earendil_Silmarils:
Les Anamorphoses:
"Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d'horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4)."
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
"… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s'opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d'établir
de façon élégante des résultats de
la géométrie élémentaire."
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
"This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them."
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
"… is nothing else but the
relativity theory for the set Sigma,
a set which, by its discrete and
finite character, is conceptually
so much simpler than the
infinite set of points in space
or space-time dealt with
by ordinary relativity theory."
— Weyl, Symmetry,
Princeton U. Press, 1952
Metaphor and Algebra…
relativity theory for the set Sigma,
a set which, by its discrete and
finite character, is conceptually
so much simpler than the
infinite set of points in space
or space-time dealt with
by ordinary relativity theory."
— Weyl, Symmetry,
Princeton U. Press, 1952
Metaphor and Algebra…
"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."
— attributed, in varying forms, to
Max Black, Models and Metaphors, 1962
Max Black, Models and Metaphors, 1962
For metaphor and
algebra combined, see
"Symmetry invariance
in a diamond ring,"
in a diamond ring,"
A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
More on Max Black…
"When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated."
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
A New Slant…
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase "4×4 square" with the phrase "projective geometry." The results are ridiculously inappropriate, but at least the second example does, literally, illuminate "new slants"– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
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Sunday, July 13, 2003
Sunday July 13, 2003
ART WARS, 5:09
The Word in the Desert
For Harrison Ford in the desert.
(See previous entry.)
Words strain,
Crack and sometimes break,
under the burden,
Under the tension, slip, slide, perish,
Will not stay still. Shrieking voices
Scolding, mocking, or merely chattering,
Always assail them.
The Word in the desert
Is most attacked by voices of temptation,
The crying shadow in the funeral dance,
The loud lament of
the disconsolate chimera.— T. S. Eliot, Four Quartets
The link to the word "devilish" in the last entry leads to one of my previous journal entries, "A Mass for Lucero," that deals with the devilishness of postmodern philosophy. To hammer this point home, here is an attack on college English departments that begins as follows:
"William Faulkner's Snopes trilogy, which recounts the generation-long rise of the drily loathsome Flem Snopes from clerk in a country store to bank president in Jefferson, Mississippi, teems with analogies to what has happened to English departments over the past thirty years."
For more, see
The Word in the Desert,
by Glenn C. Arbery.
See also the link on the word "contemptible," applied to Jacques Derrida, in my Logos and Logic page.
This leads to an National Review essay on Derrida,
The Philosopher as King,
by Mark Goldblatt.
A reader's comment on my previous entry suggests the film "Scotland, PA" as viewing related to the Derrida/Macbeth link there.
I prefer the following notice of a 7-11 death, that of a powerful art museum curator who would have been well cast as Lady Macbeth:
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Die Fahne Hoch, |
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From the Whitney Museum site:
"Max Anderson: When artist Frank Stella first showed this painting at The Museum of Modern Art in 1959, people were baffled by its austerity. Stella responded, 'What you see is what you see. Painting to me is a brush in a bucket and you put it on a surface. There is no other reality for me than that.' He wanted to create work that was methodical, intellectual, and passionless. To some, it seemed to be nothing more than a repudiation of everything that had come before—a rational system devoid of pleasure and personality. But other viewers saw that the black paintings generated an aura of mystery and solemnity.
The title of this work, Die Fahne Hoch, literally means 'The banner raised.' It comes from the marching anthem of the Nazi youth organization. Stella pointed out that the proportions of this canvas are much the same as the large flags displayed by the Nazis.
But the content of the work makes no reference to anything outside of the painting itself. The pattern was deduced from the shape of the canvas—the width of the black bands is determined by the width of the stretcher bars. The white lines that separate the broad bands of black are created by the narrow areas of unpainted canvas. Stella's black paintings greatly influenced the development of Minimalism in the 1960s."
From Play It As It Lays:
She took his hand and held it. "Why are you here."
"Because you and I, we know something. Because we've been out there where nothing is. Because I wanted—you know why."
"Lie down here," she said after a while. "Just go to sleep."
When he lay down beside her the Seconal capsules rolled on the sheet. In the bar across the road somebody punched King of the Road on the jukebox again, and there was an argument outside, and the sound of a bottle breaking. Maria held onto BZ's hand.
"Listen to that," he said. "Try to think about having enough left to break a bottle over it."
"It would be very pretty," Maria said. "Go to sleep."
I smoke old stogies I have found…
Cigar Aficionado on artist Frank Stella:
" 'Frank actually makes the moment. He captures it and helps to define it.'
This was certainly true of Stella's 1958 New York debut. Fresh out of Princeton, he came to New York and rented a former jeweler's shop on Eldridge Street on the Lower East Side. He began using ordinary house paint to paint symmetrical black stripes on canvas. Called the Black Paintings, they are credited with paving the way for the minimal art movement of the 1960s. By the fall of 1959, Dorothy Miller of The Museum of Modern Art had chosen four of the austere pictures for inclusion in a show called Sixteen Americans."
For an even more austere picture, see
For more on art, Derrida, and devilishness, see Deborah Solomon's essay in the New York Times Magazine of Sunday, June 27, 1999:
"Blame Derrida and
his fellow French theorists…."
See, too, my site
Art Wars: Geometry as Conceptual Art.
For those who prefer a more traditional meditation, I recommend
("Behold the Wood of the Cross")
For more on the word "road" in the desert, see my "Dead Poet" entry of Epiphany 2003 (Tao means road) as well as the following scholarly bibliography of road-related cultural artifacts (a surprising number of which involve Harrison Ford):