Log24

Friday, December 9, 2011

Duality

Filed under: General — m759 @ 2:45 pm

Some background for last night's post on Diego Rivera—

IMAGE- Google Doodle of an imagined Diego Rivera painting the sun in a mural

See "Octavio Paz" + "Solar Country" in this journal.

"Mexico is a solar country— but it is also a black country, a dark country.
This duality of Mexico has preoccupied me since I was a child."

Octavio Paz

Sunday, August 11, 2024

Double Duals

Filed under: General — Tags: — m759 @ 8:40 am

"That in which space itself is contained"
Wallace Stevens

In the ninefold square,
projective-perspectivity duality
corresponds to
projective-correlation duality.

Illustrations —

Wednesday, August 2, 2023

Adumbrating the Paradoxology

Filed under: General — Tags: , — m759 @ 12:04 pm

"William Blake's statement in The Marriage of Heaven and Hell
'Eternity is in love with the productions of time' is an adumbration
of the paradoxology of the game of hide-and-seek that Non-duality
is playing with and in celebration of itself in Ia divina commedia of
this night of its dream."

— Joseph Campbell in "The Inner Reaches of Outer Space" (©1986)

The above passage is from this journal on May 25 . . .
the release date for all eight episodes of FUBAR.

Thursday, May 25, 2023

Woo for Burton (Tara Isabella Burton, that is)

Filed under: General — Tags: — m759 @ 10:07 am

"William Blake's statement in The Marriage of Heaven and Hell
'Eternity is in love with the productions of time' is an adumbration
of the paradoxology of the game of hide-and-seek that Non-duality
is playing with and in celebration of itself in Ia divina commedia of
this night of its dream."

— Joseph Campbell in "The Inner Reaches of Outer Space" (©1986)

Related material from a Log24 search for "inscapes4"—

Friday, January 6, 2023

Death of a Sandal Maker

Filed under: General — m759 @ 4:36 pm

Tuesday, November 8, 2016, was also the publication date
at Princeton University Press for a book by one Herbert Gintis:

Gintis reportedly died yesterday, Jan. 5, 2023.

"He graduated from the University of Pennsylvania
and then attended Harvard University for post-graduate
work in mathematics. After receiving his master's degree,
he grew disillusioned with the field, and while at Harvard,
became a sandal maker with a shop in Harvard Square." 

In later years, Gintis was associated with the Santa Fe Institute.

Saturday, May 28, 2022

Grothendieck at Chapman …

Filed under: General — m759 @ 1:41 pm

Last two days of the conference, May 27 and 28, 2022 —

27th Friday

9:00 – 10:00 Andrés Villaveces (Univ. Nacional de Colombia):
Galoisian model theory:
the role(s) of Grothendieck (à son insu! )

10:00 – 11:00 Olivia Caramello (Univ. of Insubria; by Zoom):
The “unifying notion” of topos 1

1:00 – 11:15 Coffee Break

1:15 – 12:15 Mike Shulman (Univ. of San Diego):
Lifting Grothendieck universes to Grothendieck toposes 

12:15 – 1:15 José Gil-Ferez (Chapman Univ.)
The Isomorphism Theorem of Algebraic Logic:
a Categorical  Perspective

1:15 – 2:30 Lunch

2:30 – 3:30 Oumar Wone (Chapman) :
Vector bundles on Riemann surfaces according to
Grothendieck and his followers

3:30 – 4:30 Claudio Bartocci (Univ. of Genova):
The inception of the theory of moduli spaces:
Grothendieck's Quot scheme

4:30 – 5:30 Christian Houzel (IUFM de Paris):
Riemann surfaces after Grothendieck
[presented by J.J. Szczeciniarz]

28th Saturday

9:00 – 10:00 Silvio Ghilardi (Univ. degli Studi, Milano):
Investigating definability in propositional logic
via Grothendieck topologies and sheaves

10:00 – 11:00 Matteo Viale (Univ. of Turin; by zoom):
The duality between Boolean valuated models and 
topological presheaves

11:00 – 11:15 Coffee Break

11:15 – 12:15 Benjamin Collas (RIMS, Kyoto Univ.):
Galois-Teichmüller: arithmetic geometric principles

12:15 – 1:15 Closing: general discussion
animated by Alex Kurz (Chapman)

Saturday, January 16, 2021

Dark City . . .

Filed under: General — Tags: — m759 @ 1:28 pm

Continues .

“Dark City is an action movie — and like all good sci-fi movies,
it has aliens in it, too. The aliens have the problem that they
do not possess individual identities or souls, and for that reason,
their race is on the brink of extinction. To prevent this from
happening they perform experiments on the inhabitants of the
city to learn the secret of individuality and to eventually acquire it.
The key ingredient is memory.”

— Chapter 13 of SHELL BEACH: The search for the final theory,
by Jesper Møller Grimstrup, published on January 10, 2021.

“She did not ask herself if the Shorter Way was really there,
did not wonder if she was easing into a delusion.
The issue was settled. Here it was.”

— Joe Hill,  NOS4A2  (p. 680).  William Morrow, April 30, 2013.

A different “shorter way” —

Friday, February 24, 2017

Abstract Configurations on Good Friday

Filed under: General,Geometry — m759 @ 9:16 pm

An arXiv article from Good Friday, 2003, by Igor Dolgachev,
a student of the late Igor Shafarevich (see previous post) —

See also my Dec. 29, 1986, query on Duality and Symmetry.

Thursday, August 18, 2016

An Oxford Education:

Filed under: General — m759 @ 3:33 pm

Or, "An Education Continued"

This journal on May 29, 2010, had a followup to
the previous day's post "Multispeech for Oxford"—

An Oxford workshop, "Quantum Physics and Logic," began
on the date of the above Log24 "Packed" post, May 29, 2010.
The first talk was by John Baez —

Baez's notes on his talk begin
"Duality has many manifestations in logic and physics." —

'Duality has many manifestations in logic and physics.'- John Baez

Yes, it does.

Tuesday, December 1, 2015

Pascal’s Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 12:01 am

See a search for "large Desargues configuration" in this journal.

The 6 Jan. 2015 preprint "Danzer's Configuration Revisited," 
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the 
interesting context of Pascal's Hexagrammum Mysticum .

They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .

This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.

The duality was, as they note, previously described in 1898.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Related material on six-set geometry from a more recent source —

Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Thursday, April 19, 2012

In Memoriam

Filed under: General — m759 @ 6:48 pm

For Octavio Paz, who died on this date in 1998

… the free-standing, two-sided “Life-Death Figure,”
carved from stone in Mexico some time between
A.D. 900 and 1250, 
has multiple personalities.

— Holland Cotter,  New York Times  online today

Tuesday, December 13, 2011

Mathematics and Narrative, continued

Filed under: General,Geometry — m759 @ 11:01 pm

Mathematics —

(Some background for the Galois tesseract )

(Click to enlarge)

http://www.log24.com/log/pix11C/111213-Edge-geometry-heptads-500w.jpg

Narrative

An essay on science and philosophy in the January 2012
Notices of the American Mathematical Society .

Note particularly the narrative explanation of the double-slit experiment—

"The assertion that elementary particles have
free will and follow Quality very closely leads to
some startling consequences. For instance, the
wave-particle duality paradox, in particular the baffling
results of the famous double slit experiment,
may now be reconsidered. In that experiment, first
conducted by Thomas Young at the beginning
of the nineteenth century, a point light source
illuminated a thin plate with two adjacent parallel
slits in it. The light passing through the slits
was projected on a screen behind the plate, and a
pattern of bright and dark bands on the screen was
observed. It was precisely the interference pattern
caused by the diffraction patterns of waves passing
through adjacent holes in an obstruction. However,
when the same experiment was carried out much
later, only this time with photons being shot at
the screen one at a time—the same interference
pattern resulted! But the Metaphysics of Quality
can offer an explanation: the photons each follow
Quality in their actions, and so either individually
or en masse (i.e., from a light source) will do the
same thing, that is, create the same interference
pattern on the screen."

This is from "a Ph.D. candidate in mathematics at the University of Calgary."
His essay is titled "A Perspective on Wigner’s 'Unreasonable Effectiveness
of Mathematics.'" It might better be titled "Ineffective Metaphysics."

Saturday, November 26, 2011

Innermost Kernel (continued*)

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

A search on the word "innermost" in a PDF copy of a book
by Suzanne Gieser on Jung and Pauli yields no definite meaning
for the book's title, The Innermost Kernel  (Springer, 2005).

The author does, however, devote a section (pp. 36-41) to the
influence of Schopenhauer on Jung and Pauli, and that section at least
suggests that the historical  origin of her title is in Schopenhauer's
reformulation of Kant's "Ding an sich."

The Innermost Kernel , p. 37—

"… an expression of an underlying invisible world,
the one that forms the innermost essence of reality,
the thing-in-itself. This is the will, a blind existence
that forms an omnipresent entity beyond time, space
and individuality." *

* Arthur Schopenhauer, "Über die Vierfache Wurzel
  des Satzes vom zureichenden Grunde" (1813),
  Kleinere Schriften, SämtlicheWerke III 
  (Stuttgart, 1962), 805–806.

* See also Mann on Schopenhauer and an "innermost kernel."

Tuesday, December 14, 2010

Play and Interplay

Filed under: General,Geometry — m759 @ 4:23 am

Julie Taymor in an interview published Dec. 12 —

“I’ve got two Broadway shows, a feature film, and Mozart,’’ she said. “It’s a very interesting place to be and to be able to move back and forth, but at a certain point you have to be able to step outside and see,’’ and here she dropped her voice to a tranquil whisper, “it’s just theater. It’s all theater. It’s all theater. The whole thing is theater.’’

Non-theater —

"The interplay between Euclidean and Galois  geometry" and
related remarks on interplay — Keats's Laws of Aesthetics.

Part theater, part non-theater —

Cubist crucifixion.

Saturday, December 11, 2010

Dual Duel

Filed under: General,Geometry — m759 @ 7:20 am

"Shifting Amid, and Asserting, His Own Cinema"

— Headline of an essay on Bertolucci in tormorrow's Sunday New York Times

This, together with yesterday's post on the Paris "Symmetry, Duality, and Cinema" conference last June, suggests a review of the phrase "blue diamond" in this journal. The search shows a link to the French art film "Duelle."

Some background for the word and concept from a French dictionary

duel
  adjectif masculin singulier
1 relatif à la dualité, à ce qui est double, constitué de deux éléments distincts
  nom masculin singulier
2 combat opposant deux personnes, à l'arme blanche ou au pistolet, afin de chercher réparation d'un dommage ou d'une injure de l'un des combattants
3 par extension compétition, conflit
4  (linguistique) dans certaines langues, cas de nombre distinct du singulier et du pluriel, correspondant à une action effectuée par deux personnes

duelle
  adjectif féminin singulier
relative à la dualité, à ce qui est double, constitué de deux éléments distincts

For examples of  duel  and duelle  see Evariste Galois
and Helen Mirren (the latter in The Tempest  and in 2010 ).

IMAGE-- Imaginary movie poster- 'The Galois Connection'- from stoneship.org

Image from stoneship.org

Friday, December 10, 2010

Cruel Star, Part II

Filed under: General,Geometry — Tags: — m759 @ 2:00 pm

Symmetry, Duality, and Cinema

— Title of a Paris conference held June 17, 2010

From that conference, Edward Frenkel on symmetry and duality

"Symmetry plays an important role in geometry, number theory, and quantum physics. I will discuss the links between these areas from the vantage point of the Langlands Program. In this context 'duality' means that the same theory, or category, may be described in two radically different ways. This leads to many surprising consequences."

Related material —

http://www.log24.com/log/pix10B/101210-CruelStarPartII.jpg

See also  "Black Swan" in this journal, Ingmar Bergman's production of Yukio Mishima's "Madame de Sade," and Duality and Symmetry, 2001.

This journal on the date of the Paris conference
had a post, "Nighttown," with some remarks about
the duality of darkness and light. Its conclusion—

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

Sunday, October 10, 2010

10/10/10

Filed under: General,Geometry — m759 @ 10:10 am

Pythagoreans might regard today as the Day of the Tetraktys.

Some relevant epigraphs—

"Contrary to John Keats's First and Second Laws of Aesthetics ('Beauty is truth, truth beauty') truth and beauty are poles apart. Keats's ode itself, while denying this by precept, bears it out by example. Truth occupies the alethic pole of the intellectual sphere and beauty the aesthetic pole. Each is admirable in its way. The alethic pole exerts the main pull on science, in the broad sense: Wissenschaft,  comprising mathematics, history, and all the hard and soft sciences in between. The aesthetic pole is the focus of belles lettres,  music, art for art's sake."

— W. V. Quine in Quiddities

Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaos und Platon

Original title of Burkert's Lore and Science in Ancient Pytthagoreanism

"What song the Sirens sang…" — Sir Thomas Browne

Recommended:

  1. Stanford Encyclopedia of Philosophy  on the Pythagorean acusmatici  and mathematici
  2. Burkert on the same topic
  3. Father Robert Sokolowski's foreword to Rota's Indiscrete Thoughts

Monday, May 17, 2010

Rolling the Stone

Filed under: General,Geometry — m759 @ 10:12 pm

A new NY Times column:

http://www.log24.com/log/pix10A/100517-NYT-Stone.jpg

Today's New York Times
re-edited for philosophers:

http://www.log24.com/log/pix10A/100517-JonesClue.jpg

See also

Eightfold Symmetry,

John Baez's paper
Duality in Logic and Physics
(for a May 29 meeting at Oxford),

The Shining of May 29, and

Lubtchansky's Key, with its links
to Duelle (French, f. adj., dual)
and Art Wars for Trotsky's Birthday.

Friday, January 22, 2010

Meta Physics

Filed under: General — m759 @ 9:00 am

Church Emblem

Yin-Yang emblem of duality in Niels Bohr's coat of arms, Frederiksborg Castle Church

Related material:

Faust in Copenhagen,
Meta Physics,
and yesterday's entry.

Monday, January 4, 2010

Google’s Apple Tree

Filed under: General — m759 @ 11:30 am

http://www.log24.com/log/pix10/100104-Apple.jpg

Google has illuminated its search page today with a falling apple in honor of what it is pleased to call the birthday of Newton. (When Newton was born, the calendar showed it was Christmas Day, 1642; Google prefers to associate Sir Isaac with a later version of the calendar.)

Some related observations–

Adapted from a Log24 entry
of Monday, March 24, 2008–
 

 

"Hanging from the highest limb
of the apple tree are
     the three God's Eyes…"

    — Ken Kesey

"But what's beautiful can't be bad. You're not bad, North Wind?"

"No; I'm not bad. But sometimes beautiful things grow bad by doing bad, and it takes some time for their badness to spoil their beauty. So little boys may be mistaken if they go after things because they are beautiful."

"Well, I will go with you because you are beautiful and good, too."

"Ah, but there's another thing, Diamond:– What if I should look ugly without being bad– look ugly myself because I am making ugly things beautiful?– What then?"

"I don't quite understand you, North Wind. You tell me what then."

"Well, I will tell you. If you see me with my face all black, don't be frightened. If you see me flapping wings like a bat's, as big as the whole sky, don't be frightened. If you hear me raging ten times worse than Mrs. Bill, the blacksmith's wife– even if you see me looking in at people's windows like Mrs. Eve Dropper, the gardener's wife– you must believe that I am doing my work. Nay, Diamond, if I change into a serpent or a tiger, you must not let go your hold of me, for my hand will never change in yours if you keep a good hold. If you keep a hold, you will know who I am all the time, even when you look at me and can't see me the least like the North Wind. I may look something very awful. Do you understand?"

"Quite well," said little Diamond.

"Come along, then," said North Wind, and disappeared behind the mountain of hay.

Diamond crept out of bed and followed her.

    — George MacDonald,
      At the Back of the North Wind

   

From Log24 on Sunday, March 23, 2008–

 
A sequel to
The Crimson Passion

Easter Egg

Jill St. John with diamond

Click on image
 for further details.


Duality:


A pair of book covers in honor
  of the dies natalis of T. S. Eliot–

http://www.log24.com/log10/saved/100103-TheAristocrat_files/100104-Duality.jpg

From Virginia Woolf,  "Modern Fiction" (Ch. 13 in The Common Reader, First Series)

Woolf on what she calls "materialist" fiction–

Life escapes; and perhaps without life nothing else is worth while. It is a confession of vagueness to have to make use of such a figure as this, but we scarcely better the matter by speaking, as critics are prone to do, of reality. Admitting the vagueness which afflicts all criticism of novels, let us hazard the opinion that for us at this moment the form of fiction most in vogue more often misses than secures the thing we seek. Whether we call it life or spirit, truth or reality, this, the essential thing, has moved off, or on, and refuses to be contained any longer in such ill-fitting vestments as we provide. Nevertheless, we go on perseveringly, conscientiously, constructing our two and thirty chapters after a design which more and more ceases to resemble the vision in our minds. So much of the enormous labour of proving the solidity, the likeness to life, of the story is not merely labour thrown away but labour misplaced to the extent of obscuring and blotting out the light of the conception. The writer seems constrained, not by his own free will but by some powerful and unscrupulous tyrant who has him in thrall, to provide a plot, to provide comedy, tragedy, love interest, and an air of probability embalming the whole so impeccable that if all his figures were to come to life they would find themselves dressed down to the last button of their coats in the fashion of the hour. The tyrant is obeyed; the novel is done to a turn. But sometimes, more and more often as time goes by, we suspect a momentary doubt, a spasm of rebellion, as the pages fill themselves in the customary way. Is life like this? Must novels be like this?

Look within and life, it seems, is very far from being “like this”. Examine for a moment an ordinary mind on an ordinary day. The mind receives a myriad impressions—trivial, fantastic, evanescent, or engraved with the sharpness of steel. From all sides they come, an incessant shower of innumerable atoms; and as they fall, as they shape themselves into the life of Monday or Tuesday, the accent falls differently from of old; the moment of importance came not here but there; so that, if a writer were a free man and not a slave, if he could write what he chose, not what he must, if he could base his work upon his own feeling and not upon convention, there would be no plot, no comedy, no tragedy, no love interest or catastrophe in the accepted style, and perhaps not a single button sewn on as the Bond Street tailors would have it. Life is not a series of gig lamps symmetrically arranged; life is a luminous halo, a semi-transparent envelope surrounding us from the beginning of consciousness to the end. Is it not the task of the novelist to convey this varying, this unknown and uncircumscribed spirit, whatever aberration or complexity it may display, with as little mixture of the alien and external as possible? We are not pleading merely for courage and sincerity; we are suggesting that the proper stuff of fiction is a little other than custom would have us believe it.

It is, at any rate, in some such fashion as this that we seek to define the quality which distinguishes the work of several young writers, among whom Mr. James Joyce is the most notable….

Tuesday, April 7, 2009

Tuesday April 7, 2009

Filed under: General — m759 @ 5:24 pm

Bright Star and Dark Lady

on July 26, 2003

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

 

Dark Lady

The same story on
May 11, 2005

 with a different
dark lady:

http://www.log24.com/log/pix09/090407-SnowWhiteQueen.jpg

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General — m759 @ 4:30 am
Shine On

(Continued from
April 21, 2007)

From “Today in History,”
by the Associated Press–

April 19, 2008–
“On this date….  
Ten years ago….
Mexican poet-philosopher
Octavio Paz died at age 84.”

“Mexico is a solar country–
but it is also a black country,
a dark country. This duality
of Mexico has preoccupied
me since I was a child.”

Octavio Paz, as quoted
   by Homero Aridjis

“And the light shineth in
darkness; and the darkness
comprehended it not.”
— John 1:5  

Images of time and eternity in memory of Michelangelo

Ya la ronda
  llega aquí

Sunday, December 16, 2007

Sunday December 16, 2007

Filed under: General — Tags: , , — m759 @ 1:09 pm
 
Mad Phaedrus
Meets Mad Ezra

"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

"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

"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.
 

Related material: St. Augustine's Day 2006

Thursday, October 26, 2006

Thursday October 26, 2006

Filed under: General — m759 @ 12:00 pm
Hardy & Wright 
The image “http://www.log24.com/log/pix06A/061025-Wright.jpg” cannot be displayed, because it contains errors.

“When he was taken to church
he amused himself by factorizing
the numbers of the hymns.”

— C. P. Snow, foreword to
A Mathematician’s Apology,
by G. H. Hardy

An application of
lottery hermeneutics:

420 –> 4/20 –>

Hall of Shame,
Easter Sunday,
April 20, 2003;

145 –> 5*29 –> 5/29 –>

The Shining of May 29.

The Rev. Wright may also
be interested in the following

Related material:

“Shem was a sham….”
(FW I.7, 170 and Log24 Oct. 13),
and The Hebrew Word Shem:

“When I teach introductory Hebrew, the first word I typically teach is the common noun SHEM. It’s pronounced exactly like our English word ‘shame,’ means almost exactly the opposite, and seems to me to be a key….” — Glen Penton

This word occurs, notably, in Psalm (or “hymn”) 145.

See http://scripturetext.com/psalms/145-1.htm:

thy name
shem  (shame)
an appellation, as a mark or memorial of individuality; by implication honor, authority, character — + base, (in-)fame(-ous), named(-d), renown, report.

Update of 12:25 PM 10/26
from the online Crimson:


Related material:
The Crimson Passion

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , , — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

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

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Sunday, August 13, 2006

Sunday August 13, 2006

Filed under: General,Geometry — Tags: — m759 @ 6:00 pm
Happy Six

(continued from
New Year's Day, 2006)

See David P. Roberts (1998)
on Twin Sextic Algebras
for a discussion of
sextic twinning as an
analogue of duality
in vector spaces:

The image “http://www.log24.com/log/pix06A/060813-Twinning.jpg” cannot be displayed, because it contains errors.

Related material:

R.T. Curtis, 2001:

"A Fresh Approach
to the Exceptional Automorphism
and Covers of the Symmetric Groups"
in
The Arabian Journal
for Science and Engineering
.
 

Saturday, June 25, 2005

Saturday June 25, 2005

Filed under: General — Tags: — m759 @ 12:00 am
Merry AntiChristmas!

Religious Symbolism
at Midnight:

The image “http://www.log24.com/log/pix05A/050625-Star.jpg” cannot be displayed, because it contains errors.

Related material:

Star Wars 6/13/05,
Dark City 6/14/05,
and De Arco, as well
as the following from
July 26, 2003:

Bright Star and Dark Lady

"Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child."

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

 

Dark Lady

Tuesday, May 24, 2005

Tuesday May 24, 2005

Filed under: General — m759 @ 2:00 pm
Final Arrangements, continued:

Two Poles

From today’s New York Times:

The image “http://www.log24.com/log/pix05/050524-NYT.jpg” cannot be displayed, because it contains errors.

From erraticimpact.com on Paul Ricoeur:

“Ricoeur reserves his greatest admiration for
the narratologist Algirdas-Julien Greimas.
[See below.]
Ricoeur also explores the relationship
between the philosophical and religious
domains, attempting to reconcile
the two poles in his thought.”

From today’s NYT obituary of Sol Stetin:

“Mr. Stetin, who emigrated from Poland at the age of 10 and dropped out of high school in the ninth grade, was fond of saying he got his education in the labor movement.”

The image “http://www.log24.com/log/pix05/050524-JP2.jpg” cannot be displayed, because it contains errors.


“… 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

Related material:

Poetry’s Bones and
Theme and Variations.

Other readings on polarity:

Log24, May 24, 2003, and
from July 26, 2003:

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

Dark Lady

Tuesday, April 6, 2004

Tuesday April 6, 2004

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

Ideas and Art, Part III

The first idea was not our own.  Adam
In Eden was the father of Descartes…

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"Quaedam ex his tanquam rerum imagines sunt, quibus solis proprie convenit ideae nomen: ut cùm hominem, vel Chimaeram, vel Coelum, vel Angelum, vel Deum cogito."

Descartes, Meditationes III, 5

"Of my thoughts some are, as it were, images of things, and to these alone properly belongs the name idea; as when I think [represent to my mind] a man, a chimera, the sky, an angel or God."

Descartes, Meditations III, 5

Begin, ephebe, by perceiving the idea
Of this invention, this invented world,
The inconceivable idea of the sun.

You must become an ignorant man again
And see the sun again with an ignorant eye
And see it clearly in the idea of it.

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"… Quinimo in multis saepe magnum discrimen videor deprehendisse: ut, exempli causâ, duas diversas solis ideas apud me invenio, unam tanquam a sensibus haustam, & quae maxime inter illas quas adventitias existimo est recensenda, per quam mihi valde parvus apparet, aliam verò ex rationibus Astronomiae desumptam, hoc est ex notionibus quibusdam mihi innatis elicitam, vel quocumque alio modo a me factam, per quam aliquoties major quàm terra exhibetur; utraque profecto similis eidem soli extra me existenti esse non potest, & ratio persuadet illam ei maxime esse dissimilem, quae quàm proxime ab ipso videtur emanasse."

Descartes, Meditationes III, 11

"… I have observed, in a number of instances, that there was a great difference between the object and its idea. Thus, for example, I find in my mind two wholly diverse ideas of the sun; the one, by which it appears to me extremely small draws its origin from the senses, and should be placed in the class of adventitious ideas; the other, by which it seems to be many times larger than the whole earth, is taken up on astronomical grounds, that is, elicited from certain notions born with me, or is framed by myself in some other manner. These two ideas cannot certainly both resemble the same sun; and reason teaches me that the one which seems to have immediately emanated from it is the most unlike."

Descartes, Meditations III, 11

"Et quamvis forte una idea ex aliâ nasci possit, non tamen hîc datur progressus in infinitum, sed tandem ad aliquam primam debet deveniri, cujus causa sit in star archetypi, in quo omnis realitas formaliter contineatur, quae est in ideâ tantùm objective."

Descartes, Meditationes III, 15

"And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality [or perfection] that is found objectively [or by representation] in these ideas is contained formally [and in act]."

Descartes, Meditations III, 15

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction,"

The Quest for the Fiction of the Absolute:

"Canto nine considers the movement of the poem between the particular and the general, the immanent and the transcendent: "The poem goes from the poet's gibberish to / The gibberish of the vulgate and back again. / Does it move to and fro or is it of both / At once?" The poet, the creator-figure, the shadowy god-figure, is elided, evading us, "as in a senseless element."  The poet seeks to find the transcendent in the immanent, the general in the particular, trying "by a peculiar speech to speak / The peculiar potency of the general." In playing on the senses of "peculiar" as particular and strange or uncanny, these lines play on the mystical relation of one and many, of concrete and abstract."

Brian Cronin in Foundations of Philosophy:

"The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight."

— From Ch. 2, Identifying Direct Insights

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction":

"The fourth canto returns to the theme of opposites. 'Two things of opposite natures seem to depend / On one another . . . . / This is the origin of change.'  Change resulting from a meeting of opposities is at the root of Taoism: 'Tao produced the One. / The One produced the two. / The two produced the three. / And the three produced the ten thousand things' (Tao Te Ching 42) …."

From an entry of March 7, 2004

 

From the web page

Introduction to the I Ching–
By Richard Wilhelm
:

"He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the 'great primal beginning' of all that exists, t'ai chi — in its original meaning, the 'ridgepole.' Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin,

.

This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites."

The t'ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that

"W.J.T. Mitchell calls 'iconology'
a study of the 'logos'
(the words, ideas, discourse, or 'science')
of 'icons' (images, pictures, or likenesses).
It is thus a 'rhetoric of images'
(Iconology: Image, Text, Ideology, p. 1)."

A variation on the t'ai chi symbol appears in a log24.net entry for March 5:

The Line,
by S. H. Cullinane

See too my web page Logos and Logic, which has the following:

"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi, was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies, ed. Quarto, Gallimard, 1999, p. 100

 Logos Alogos,
by S. H. Cullinane 

In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day
at the Sorbonne.
We shall return at twilight
from the lecture
Pleased that
the irrational is rational…."

This is the logoi alogoi of Simone Weil.

In "Notes toward a Supreme Fiction,"
Wallace Stevens lists three criteria
for a work of the imagination:

It Must Be Abstract

The Line,
by S.H. Cullinane 

It Must Change

 The 24,
by S. H. Cullinane

It Must Give Pleasure

Puzzle,
by S. H. Cullinane

Related material:

Logos and Logic.

 

Sunday, March 7, 2004

Sunday March 7, 2004

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

Ridgepole

CBS News Sunday Morning today had a ridgepole ceremony for a house that was moved from China to Salem, Massachusetts.

From the web page

Introduction to the I Ching–
By Richard Wilhelm
:

"He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the 'great primal beginning' of all that exists, t'ai chi — in its original meaning, the 'ridgepole.' Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin, .

This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites."

The t'ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that 

"W.J.T. Mitchell calls 'iconology' a study of the 'logos' (the words, ideas, discourse, or 'science') of 'icons' (images, pictures, or likenesses). It is thus a 'rhetoric of images' (Iconology: Image, Text, Ideology, p. 1)."

A variation on the t'ai chi symbol appears in a log24.net entry for March 5:

The Line,
by S. H. Cullinane

See too my web page Logos and Logic, which has the following:

"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi, was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies, éd. Quarto, Gallimard, 1999, p. 100

Logos Alogos,
 by S. H. Cullinane 

In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day
     at the Sorbonne.
 We shall return at twilight
     from the lecture
 Pleased that
     the irrational is rational…."

This is the logoi alogoi of Simone Weil.

Thursday, February 5, 2004

Thursday February 5, 2004

Filed under: General — m759 @ 12:00 pm

Affirmation of Place and Time:
East Coker and Grand Rapids

This morning’s meditation:

“Let us talk together with the courage, humor, and ardor of Socrates.

In that long conversation, we may find ourselves considering something Plato’s follower Plotinus said long ago about ‘a principle which transcends being,’ in whose domain one can ‘assert identity without the affirmation of being.’  There, ‘everything has taken its stand forever, an identity well pleased, we might say, to be as it is…. Its entire content is simultaneously present in that identity: this is pure being in eternal actuality; nowhere is there any future, for every then is a now; nor is there any past, for nothing there has ever ceased to be.’  Individuality and existence in space and time may be masks that our sensibilities impose on the far different face of quantum reality.”

— Peter Pesic, Seeing Double: Shared Identities in Physics, Philosophy, and Literature, MIT Press paperback, 2003, p. 145

A search for more on Plotinus led to sites on the Trinity, which in turn led to the excellent archives at Calvin College in Grand Rapids.

A search for the theological underpinnings of Calvin College led to the Christian Reformed church:

“Our emblem is
the cross in a triangle.”

The triangle, as a symbol of “the delta factor,” also plays an important role in the semiotic theory of Walker Percy.  A search for current material on Percy led back to one of my favorite websites, that of Percy expert Karey Perkins, and thus to the following paper:

The “East Coker” Dance
in T. S. Eliot’s Four Quartets:
An Affirmation of Place and Time

by Karey Perkins

For a rather different, but excellent, literary affirmation of place and time — in Grand Rapids, rather than East Coker — see, for instance, Michigan Roll, a novel by Tom Kakonis.

We may, for the purposes of this trinitarian meditation, regard Percy and Kakonis as speaking for the Son and Karey Perkins as a spokesperson for the Holy Spirit.  As often in my meditations, I choose to regard the poet Wallace Stevens as speaking perceptively about (if not for, or as) the Father.  A search for related material leads to a 1948 comment by Thomas McGreevy, who

“… wrote of Stevens’ ‘Credences of Summer’ (Collected Poems 376),

On every page I find things that content me, as ‘The trumpet of the morning blows in the clouds and through / The sky.’

A devout Roman Catholic, he added, ‘And I think my delight in it is of the Holy Spirit.’ (26 May 1948).”

An ensuing search for material on “Credences of Summer” led back, surprisingly, to an essay — not very scholarly, but interesting — on Stevens, Plotinus, and neoplatonism.

Thus the circle closed.

As previous entries have indicated, I have little respect for Christianity as a religion, since Christians are, in my experience, for the most part, damned liars.  The Trinity as philosophical poetry, is, however, another matter.  I respect Pesic’s speculations on identity, but wish he had a firmer grasp of his subject’s roots in trinitarian thought.  For Stevens, Percy, and Perkins, I have more than respect.

Sunday, October 26, 2003

Sunday October 26, 2003

Filed under: General — m759 @ 3:17 am

ART WARS for

Trotsky’s Birthday

Part I:
Symbols

From my entry of July 26, 2003, in memory
of Marathon Man director John Schlesinger:

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

Dark Lady

For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.

No se puede vivir sin amar.


Part II:
Sunday in the Park with Death

  To Leon from Diego —
Details of a mural,
A Dream of a Sunday Afternoon
in Alameda Park,
Fresco, 1947-48,
Alameda Hotel, Mexico City:

Three’s a Crowd:

Symbol:


Tuesday, September 2, 2003

Tuesday September 2, 2003

Filed under: General,Geometry — Tags: , — m759 @ 1:11 pm

One Ring to Rule Them All

In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.

Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:

“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’  Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”

Leonard Gillman: An Interview

This clears up a question I asked earlier in this journal….

Wednesday, May 14, 2003

Common Sense

On the mathematician Kolmogorov:

“It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.”

Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses.  Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case.  A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15.  This is cited by Gillman and Jerison in the classic Rings of Continuous Functions.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area.  See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here  (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”).

Any other references to Kolmogorov’s work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone.  I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

A response by Richard Cudney:

“In regard to your entry, it is largely correct.  The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works.  So, I suppose my entry was unfair to Gelfand.  You’re right, the issue of credit is a bit touchy since Gelfand was his student.  In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student.  At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. 

And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst.  In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A.  That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up.

This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras.  In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique.  That is to say, this whole vein of Gelfand’s work started in this joint paper.

Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge.  Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third).

As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin.  The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.”

Posted 5/14/2003 at 8:44 PM by rcudney

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General — Tags: — m759 @ 5:29 am

Funeral March

John Schlesinger dead at 77;
‘Midnight Cowboy’ director

 
Anthony Breznican
Associated Press
Jul. 26, 2003 12:00 AM

LOS ANGELES – Oscar-winning director John Schlesinger, who daringly brought gay characters into mainstream cinema with Midnight Cowboy and tapped into nightmares with the teeth-drilling torture of Marathon Man, died Friday at 77.

The British-born filmmaker…. died about 5:30 a.m….

Schlesinger also directed The Day of the Locust, based on a novel by Nathanael West.

See Heaven, Hell, and Hollywood and

Dogma Part II: Amores Perros.

From the latter:

“Then you know your body’s sent,
Don’t care if you don’t pay rent,
Sky is high and so am I,
If you’re a viper — a vi-paah.”

The Day of the Locust,
    by Nathanael West (1939),
    New Directions paperback,
    1969, page 162

This song may be downloaded at

Pot Culture, 1910-1960.

That same site begins with a traditional Mexican song…

La cucaracha, la cucaracha,
 ya no puede caminar,
 porque no quiere,
 porque le falta
 marihuana que fumar.
” 

(“The cockroach, the cockroach,
 can’t walk anymore,
 because he doesn’t want to,
 because he has no
 marihuana to smoke.”)

This suggests an appropriate funeral march for John Schlesinger:

“Ya murió la cucaracha, ya la llevan a enterrar…”La Cucaracha

Those attending Schlesinger’s wake, as opposed to his funeral, may wish to perform other numbers from the Pot Culture page, which offers a variety of “viper” songs.

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

 

Dark Lady

For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.

No se puede vivir sin amar.

Concluding Unscientific Postscript:

Oh, yes… the question of
Heaven or Hell for John Schlesinger… 

Recall that he also directed the delightful
Cold Comfort Farm and see
last year’s entry for this date.

Tuesday, June 17, 2003

Tuesday June 17, 2003

Filed under: General — Tags: , — m759 @ 5:20 pm

Claves Regni Caelorum

On actor Gregory Peck, who died Thursday, June 12, 2003:

"He had early success in 'The Keys of the Kingdom,' in which he played a priest."

As Peck noted in a videotape played at his memorial service June 16,

"As a professional," he added, "I think I'd like to be thought of as a good storyteller; that's what's always interested me."

June 16, besides being the day of Peck's memorial, was also Bloomsday.  My entry for 1 PM on Bloomsday, a day celebrating the Ulysses of James Joyce, consists of the three words "Hickory, Dickory, Dock."  A comment on that entry:

"I prefer the Wake."

The following, from the Discordian Scriptures, provides a connection between the Bloomsday mouse and the Wake of patriarch Gregory Peck.

Hickory Dickory Dock

Hickory, dickory, dock!

Here we are on higher ground at once. The clock symbolizes the spinal column, or if you prefer it, Time, chosen as one of the conditions of normal consciousness. The mouse is the Ego; "Mus", a mouse, being only "Sum", "I am", spelt Qabalistically backwards.  This Ego or Prana or Kundalini force being driven up the spine, the clock strikes one, that is, the duality of consciousness is abolished. And the force again subsides to its original level. "Hickory, dickory, dock!" is perhaps the mantra which was used by the adept who constructed this rime, thereby hoping to fix it in the minds of men; so that they might attain to Samadhi by the same method. Others attribute to it a more profound significance — which is impossible to go into at this moment, for we must turn to:
 
Humpty Dumpty sat on a wall….

The Bloom of Ulysses has a certain philosophical kinship with Yale literary critic Harold Bloom.  For material related to the latter Bloom's study of Gnosticism, see Chaos Matrix.  For the conflict between Gnostic and Petrine approaches to religion, see Poul Anderson's Operation Chaos.

From an account of Peck's memorial service:

"Mourners included… Piper Laurie…."

OK, he's in.

 

Wednesday, May 14, 2003

Wednesday May 14, 2003

Filed under: General,Geometry — Tags: — m759 @ 2:00 pm

Common Sense

On the mathematician Kolmogorov:

“It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.”

Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses.  Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case.  A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15.  This is cited by Gillman and Jerison in the classic Rings of Continuous Functions.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area.  See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here  (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”).

Any other references to Kolmogorov’s work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone.  I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

Saturday, April 19, 2003

Saturday April 19, 2003

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

Harrowing

In memory of the many who have died on April 19, most notably Octavio Paz.

"There is a suggestion of Christ descending into the abyss for the harrowing of Hell.  But it is the Consul whom we think of here, rather than of Christ."

— Introduction to Malcolm Lowry's classic novel Under the Volcano, by Stephen Spender

"Hey, big Spender, spend a little time
with me." — Song lyric

For a somewhat deeper meditation on time, see Architecture of Eternity.

See also Literature of the Descent into Hell

"Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child."

Octavio Paz, quoted by Homero Aridjis

Amen.

Concluding Unscientific Postscripts:

"Once upon a time…" — Anonymous

"It's quarter to three…" — Sinatra

Tuesday, January 28, 2003

Tuesday January 28, 2003

Filed under: General — m759 @ 7:11 pm

State of the Communion

Relevant readings:

  • Definition of the communion of saints in the Catholic Encylopedia

    “In that communion there is no loss of individuality, yet such an interdependence that the saints are ‘members one of another’ (Rom., xii, 5)….”

  • Ephesians 4:16
  • “…the whole body fitly joined together and compacted by that which every joint supplieth….”

  • A Game Designer’s Holy Grail —

    “Herman Melville described the exact process beautifully in his novel, Mardi:

    ‘In me, many worthies recline, and converse. I list to St. Paul who argues the doubts of Montaigne…'”

  • Invitation to the HipBone Games —

    “…man is seen as he is sub specie aeternitatis, an ‘immortal diamond.'”

  • Your Hip Bone Connected —

    “Now hear the word of the Lord” 

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