A search for "Max Black" in this journal yields some images
from a post of August 30, 2006 . . .
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"Jackson has identified the seventh symbol." |
The "Jackson" above is played by the young James Spader,
who in an older version currently stars in "The Blacklist."
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"… the memorable models of science are 'speculative instruments,' — Max Black in Models and Metaphors , Cornell U. Press, 1962 |
Structure vs. Character continued…
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Structure |
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Related vocabulary:
Nick Tosches on the German word “Quell “
* The title is from Heidegger.
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195-196:
"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."
The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
Background— August 30, 2006—
In the 2006 post, the above seventh symbol 110000 was
interpreted as the I Ching hexagram with topmost and
next-to-top lines solid, not broken— Hexagram 20, View .
In a different interpretation, 110000 is the binary for the decimal
number 48— representing the I Ching's Hexagram 48, The Well .
“… Max Black, the Cornell philosopher, and
others have pointed out how ‘perhaps every science
must start with metaphor and end with algebra, and
perhaps without the metaphor there would never
have been any algebra’ ….”
– Max Black, Models and Metaphors,
Cornell U. Press, 1962, page 242, as quoted
in Dramas, Fields, and Metaphors,
by Victor Witter Turner, Cornell U. Press,
paperback, 1975, page 25
The algebra is certainly clearer than either I Ching
metaphor, but is in some respects less interesting.
For a post that combines both the above I Ching
metaphors, View and Well , see Dec. 14, 2007.
In memory of scholar Elinor Ostrom,
who died today—
David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —
See also this morning's post as well as
Monday's post quoting George David Birkhoff —
"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."
Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).
The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…
See also Geometry of the I Ching and a search in this journal for
* February 27 and March 13
** G20160 in Mitchell 1910, LF(3,22) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,22),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)
"For further details,
click on the well."
"The kind of movement Olson urges is
an inward deepening of the image,
an in-sighting of the superimposed
levels of significance within it.
This is the very mode that Jung
suggested for grasping dreams–
not as a sequence in time,
but as revolving around
a nodal complex."
And from Feb. 29, 2008:
and the following day:
— Heraclitus
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
A Cartoon Graveyard
Click to enlarge.
From Fathers' Day Meditation:
For further details,
click on the well.
Click image to enlarge.
The “Boy’s Life” illustration is of an Arthur C. Clarke story, “Against the Fall of Night.” This, according to the review quoted below, was Clarke’s first story, begun in 1936 and first published in 1948. The title is from a poem by
From a book review by Christopher B. Jones:
“Against the Fall of Night describes well how it often takes youth to bring forth change. The older mind becomes locked in a routine, or blocks out things because it has been told that it shouldn’t think or talk about them. But the young mind is ever the explorer, seeking out knowledge without the taboos placed on it by a rigid society. Alvin is a breath of fresh air in the don’t-look-over-the-wall society of Diaspar.
Myths play a big role, and an interesting religious overtone pervades the story with a long since departed being whose origins are unknown and who played an important part in Earth’s past. Parallels to Jesus can easily be drawn, and the forecast shown for the longevity of religions in general seems to me to be rather accurate….
Finally, when Alvin uncovers part of the truth he has been looking for, he learns of the dangers and stagnation that can befall a xenophobic society. There are still a few such societies in the world today, and this characteristic almost always comes with negative effects– even if it has been cultivated with the intention to protect.”
An example of such a xenophobic society is furnished by the Hadassah ad currently running in the New York Times obituaries section: “Who will say Kaddish in Israel?”
Another example:
Tom Stoppard, in the London Times of Sunday, March 16, 2008, on the social unrest of forty years ago in 1968–
“Altering the psyche was supposed to change the social structure but, as a Marxist, Max knows it really works the other way: changing the social structure is the only way to change the psyche. The idea that ‘make love, not war’ is a more practical slogan than ‘workers of the world unite’ is as airy-fairy as the I Ching.”
Airy-fairy, Jewey-phooey.
Clarke’s 1948 story was the basis of his 1956 novel, The City and the Stars. In memory of the star Richard Widmark, here are two illustrations from St. Mark’s Day, 2003:
Housman asks the reader
to tell him of runes to grave
or bastions to design
“against the fall of night.”
Here, as examples, are
one rune and one bastion.
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Represents |
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Neither part of this memorial suits the xenophobic outlook of Israel. Both parts, together, along with his classic film “The Long Ships,” seem somehow suited to the non-xenophobic outlook of Richard Widmark. As for the I Ching… perhaps Widmark has further voyages to make.
"Plato's Good was a fixed and eternal and unmoving Idea, whereas for the rhetoricians it was not an Idea at all. The Good was not a form of reality. It was reality itself, ever changing, ultimately unknowable in any kind of fixed, rigid way." –Phaedrus in Zen and the Art of Motorcycle Maintenance
This apparent conflict between eternity and time, fixity and motion, permanence and change, is resolved by the philosophy of the I Ching and by the Imagism of Ezra Pound. Consider, for example, the image of The Well
as discussed here on All Saints' Day 2003 and in the previous entry.
As background, consider the following remarks of James Hillman in "Egalitarian Typologies Versus the Perception of the Unique," Part III: Persons as Images—
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"To conceive images as static is to forget that they are numens that move. Charles Olson, a later poet in this tradition, said: 'One perception must immediately and directly lead to a further perception… always, always one perception must must must move instanter, on another.' 80 Remember Lavater and his insistence on instantaneity for reading the facial image. This is a kind of movement that is not narrational, and the Imagists had no place for narrative. 'Indeed the great poems to come after the Imagist period– Eliot’s The Waste Land and Four Quartets; Pound’s Cantos; Williams’s Paterson– contain no defining narrative.' 81 The kind of movement Olson urges is an inward deepening of the image, an in-sighting of the superimposed levels of significance within it. 82 This is the very mode that Jung suggested for grasping dreams– not as a sequence in time, but as revolving around a nodal complex. If dreams, then why not the dreamers. We too are not only a sequence in time, a process of individuation. We are also each an image of individuality."
80 The New American Poetry (D. M. Allen, ed.) N.Y.: Evergreen, Grove, 1960, pp. 387-88. from Jones, p. 42. 81 Jones,* p. 40. 82 H. D. later turned narration itself into image by writing a novel in which the stories were "compounded like faces seen one on top of another," or as she says "superimposed on one another like a stack of photographic negatives" (Jones, p. 42). Cf. Berry,** p. 63: "An image is simultaneous. No part precedes or causes another part, although all parts are involved with each other… We might imagine the dream as a series of superimpositions, each event adding texture and thickening to the rest." * Imagist Poetry (Peter Jones, ed.) London: Penguin, 1972 ** The contrast between image simultaneity and narrative succession, and the different psychological effects of the two modes, is developed by Patricia Berry, "An Approach to the Dream," Spring 1974 (N. Y./Zürich: Spring Publ.), pp. 63, 68-71 |
Hillman also says that
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"Jung’s 'complex' and Pound's definition of Image and Lavater's 'whole heap of images, thoughts, sensations, all at once' are all remarkably similar. Pound calls an Image, 'that which presents an intellectual and emotional complex in an instant of time'… 'the Image is more than an Idea. It is a vortex or cluster of fused ideas and is endowed with energy'… 'a Vortex, from which and through which, and into which, ideas are constantly rushing.' 79 Thus the movement, the dynamics, are within the complex and not only between complexes, as tensions of opposites told about in narrational sequences, stories that require arbitrary syntactical connectives which are unnecessary for reading an image where all is given at once."
79 These definitions of Image by Pound come from his various writings and can all be found in Jones, pp. 32-41. Further on complex and image, see J. B. Harmer, Victory in Limbo: Imagism 1908-17, London: Secker & Warburg, 1975, pp. 164-68. |
These remarks may help the reader to identify with Ada during her well-viewing in Cold Mountain (previous entry):
"She was dazzled by light and shade, by the confusing duplication of reflections and of frames. All coming from too many directions for the mind to take account of. The various images bounced against each other until she felt a desperate vertigo…."
If such complexity can be suggested by Hexagram 48, The Well, alone, consider the effect of the "cluster of fused ideas… endowed with energy" that is the entire 64-hexagram I Ching.
“Well, it changes.”
A related Log24 link from
that same date, November 27:
“Plato hadn’t tried to destroy areté. He had encapsulated it; made a permanent, fixed Idea out of it; had converted it to a rigid, immobile Immortal Truth. He made areté the Good, the highest form, the highest Idea of all. It was subordinate only to Truth itself, in a synthesis of all that had gone before.That was why the Quality that Phaedrus had arrived at in the classroom had seemed so close to Plato’s Good. Plato’s Good was taken from the rhetoricians. Phaedrus searched, but could find no previous cosmologists who had talked about the Good. That was from the Sophists. The difference was that Plato’s Good was a fixed and eternal and unmoving Idea, whereas for the rhetoricians it was not an Idea at all. The Good was not a form of reality. It was reality itself, ever changing, ultimately unknowable in any kind of fixed, rigid way.”
— as well as Cold Mountain —
“So in short order Ada found herself bent backward over the mossy well lip, canted in a pose with little to recommend it in the way of dignity or comfort, back arched, hips forward, legs spraddled for balance. She held a hand mirror above her face, angled to catch the surface of the water below.
Ada had agreed to the well-viewing as a variety of experiment in local custom and as a tonic for her gloom. Her thoughts had been broody and morbid and excessively retrospective for so long that she welcomed the chance to run counter to that flow, to cast forward and think about the future, even though she expected to see nothing but water at the bottom of the well.
She shifted her feet to find better grip on the packed dirt of the yard and then tried to look into the mirror. The white sky above was skimmed over with backlit haze, bright as a pearl or as a silver mirror itself. The dark foliage of oaks all around the edges framed the sky, duplicating the wooden frame of the mirror into which Ada peered, examining its picture of the well depths behind her to see what might lie ahead in her life. The bright round of well water at the end of the black shaft was another mirror. It cast back the shine of sky and was furred around the edges here and there with sprigs of fern growing between stones.
Ada tried to focus her attention on the hand mirror, but the bright sky beyond kept drawing her eye away. She was dazzled by light and shade, by the confusing duplication of reflections and of frames. All coming from too many directions for the mind to take account of. The various images bounced against each other until she felt a desperate vertigo, as if she could at any moment pitch backward and plunge head first down the well shaft and drown there, the sky far above her, her last vision but a bright circle set in the dark, no bigger than a full moon.
Her head spun and she reached with her free hand and held to the stonework of the well. And then just for a moment things steadied, and there indeed seemed to be a picture in the mirror.”
— and Log24 on December 3 —
Logo design by Anton Stankowski
"… at the beginning of the thirties… Stankowski began to work as a typographer and graphic designer in a Zurich advertising agency. Together with a group of friends– they were later to be known as the 'Zurich Concretists'– he explored the possibilities of symmetry and mirroring in the graphic arts. Stankowski experimented with squares and diagonals, making them the hallmarks of his art. Of his now world-famous logo for the Deutsche Bank— the soaring diagonal in the stable square– he proudly said in 1974: 'The company logo is a trade-mark that sends out a signal.'"
New York firefighters
killed at Deutsche Bank
"Two New York fire fighters were killed while trying to douse a blaze in the former Deutsche Bank building in the city.
The fire broke out on 14th and 15th floors yesterday afternoon and spread to several floors before it was brought under control about five hours later.
The building had been heavily damaged during the 11 September, 2001 terrorist attacks.
The building, which was damaged by falling debris of the twin towers that had collapsed in 2001 when terrorists flew hijacked planes into them, was being 'deconstructed' to make way for construction of a new Freedom Tower."
Related material
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SPORTS OF THE TIMES
Restoring the Faith
By SELENA ROBERTS
What good is a nadir if it's denied or ignored? What's the value of reaching the lowest of the low if it can't buy a cheap epiphany? |
When you care enough
to send the very best…
"Summer Reading,"
by Joost Swarte
Cover art by Barclay Shaw reprinted
from an earlier (1984) edition
Philip K. Dick as a
window wraith (see below)
"… the kind of guy who can't drink one cup of coffee without drinking six, and then stays up all night to tell you what Schopenhauer really said and how it affects your understanding of Hitchcock and what that had to do with Christopher Marlowe."
— as well as by the illustrations of Gopnik's characterization in Kernel of Eternity, and by the following passage from Gopnik's 2005 novel The King in the Window:
"What's a window wraith?"
"It's someone who once lived in the ordinary world who lives now in a window, and makes reflections of the people who pass by and look in."
"You mean you are a ghost?!" Oliver asked, suddenly feeling a little terrified.
"Just the opposite, actually. You see, ghosts come from another world and haunt you, but window wraiths are the world. We're the memory of the world. We're here for good. You're the ones who come and go like ghosts. You haunt us."
Related material: As noted, Kernel of Eternity, and also John Tierney's piece on simulated reality in last night's online New York Times. Whether our everyday reality is merely a simulation has long been a theme (as in Dick's novel above) of speculative fiction. Interest in this theme is widespread, perhaps partly because we do exist as simulations– in the minds of other people. These simulations may be accurate or may be– as is perhaps Gopnik's characterization of Philip K. Dick– inaccurate. The accuracy of the simulations is seldom of interest to the simulator, but often of considerable interest to the simulatee.
The cover of the Aug. 20 New Yorker in which the Adam Gopnik essay appears may also be of interest, in view of the material on diagonals in the Log24 entries of Aug. 1 linked to in yesterday's entry:
"Summer Reading,"
by Joost Swarte
and a Finite Model
Notes by Steven H. Cullinane
May 28, 2007
Part I: A Model of Space-Time
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Click on picture to enlarge.
Part II: A Corresponding Finite Model
The Klein quadric also occurs in a finite model of projective 5-space. See a 1910 paper:
G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.
Conwell discusses the quadric, and the related Klein correspondence, in detail. This is noted in a more recent paper by Philippe Cara:
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Related material:
The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.
The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.
There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group AGL(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.
The Freshmen, Part II
From the Daily Princetonian,
Feb. 3, 2004:
Caption: Cate Edwards' Princeton friends support her and her father.
"… when Sen. John Edwards, father of Cate Edwards '04, decided to run for president, the troop of 17 students sacrificed tans and theses to pile into a fleet of minivans headed to New Hampshire….
These volunteers… were on a first name basis with the man who had helped them move into freshman dorm rooms and had discussed Senate votes with them over Chinese food."
From Chuck Polisher's
I Ching Lexicon:
"It's claimed that
if you take a mirror
and look backwards
into a well,
you'll see your future
down in the water."
— Cold Mountain,
Vintage paperback, 1998,
page 48
"Goin' to Carolina in my mind…"
— James Taylor
A Form
John Leonard in the June 10, 2004, New York Review of Books, on E. L. Doctorow:
"… he's got urgent things to say and seeks some form to say them in, or a form that will tease and torture secret meanings out of what he thinks he already knows, or a form, like a wishing well, down which to dream, scream, or drown."
The Well. The town may be changed,
But the well cannot be changed.
It neither decreases nor increases.
They come and go and draw from the well.
If one gets down almost to the water
And the rope does not go all the way,
Or the jug breaks, it brings misfortune.From the Book of Ecclesiastes 12:6—
or ever the silver cord be loosed, or the golden bowl be broken, or the pitcher be broken at the fountain, or the wheel broken at the cistern
From Chuck Polisher's I Ching Lexicon:
See also the following form, discussed in
Balanchine's Birthday
(1/9/03) and in
Art Theory
for Yom Kippur
(10/5/03)
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