Log24

Wednesday, August 31, 2011

Odin’s Day

Filed under: Uncategorized — m759 @ 12:00 PM

Today is Wednesday.

O.E. Wodnesdæg  "Woden's day," a Gmc. loan-translation of L. dies Mercurii  "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin  . — Online Etymology Dictionary

http://www.log24.com/log/pix11B/110831-HopkinsAsOdin.jpg

Above: Anthony Hopkins as Odin in the 2011 film "Thor"

Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—

"The Tesseract* was the jewel of Odin's treasure room."

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

Weaving also played Agent Smith in The Matrix Trilogy.

The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.

"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."

— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com

For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—

http://www.log24.com/log/pix11B/110831-BrightStar.jpg

Today's New York Times  has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—

"He treated it like a puzzle."

Sometimes that's the best you can do.

* See also tesseract  in this journal.

** For a different arrangement of 13 things, see the cube's 13 axes in this journal.

Tuesday, August 30, 2011

Boundary

Filed under: Uncategorized — Tags: — m759 @ 11:07 AM

A comment yesterday on the New York Times  philosophy column “The Stone” quoted Karl Barth—

Man is the creature of the boundary between heaven and earth.”

See also Plato’s theory of ideas (or “forms”) and the I Ching

The eight trigrams are images not so much of objects as of states of change. This view is associated with the concept expressed in the teachings of Lao-tse, as also in those of Confucius, that every event in the visible world is the effect of an “image,” that is, of an idea in the unseen world. Accordingly, everything that happens on earth is only a reproduction, as it were, of an event in a world beyond our sense perception; as regards its occurrence in time, it is later than the suprasensible event. The holy men and sages, who are in contact with those higher spheres, have access to these ideas through direct intuition and are therefore able to intervene decisively in events in the world. Thus man is linked with heaven, the suprasensible world of ideas, and with earth, the material world of visible things, to form with these a trinity of the primal powers.

— Richard Wilhelm, Introduction to the I Ching

Monday, August 29, 2011

Many = Six.

Filed under: Uncategorized — m759 @ 7:20 PM

A comment today on yesterday's New York Times  philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."

Yesterday's post here  for the Feast of St. Augustine discussed the 2×2×2 cube.

Today's Augustine comment in the Times  reflects (through a glass darkly)
a Log24 post  from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.

For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Click image for some background.

Sunday, August 28, 2011

The Cosmic Part

Filed under: Uncategorized — Tags: — m759 @ 6:29 PM

Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."

A simpler candidate for the "Cube" part of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."

Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik's figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine's Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Saturday, August 27, 2011

Cosmic Cube*

Filed under: Uncategorized — Tags: , — m759 @ 12:25 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument  by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

This Way to the Egress*

Filed under: Uncategorized — m759 @ 12:00 AM

From http://msa-x.msa-x.org/?p=1064

"Exit Art New York, The Labyrinth Wall:
From Mythology to Reality" —

http://www.log24.com/log/pix11B/110827-ExitArt-LabyrinthWall.jpg

From tonight's online New York Times  obituaries —

http://www.log24.com/log/pix11B/110827-ExitArt-Ingberman.jpg

Ms. Ingberman died Wednesday. Related material—

Symmetry (Wednesday), Design (Thursday), Solomon's Labyrinth (Friday).

See also an essay by John Haber —

"Exit Art may yet offer an alternative: shut them up in the labyrinth, with the Minotaur and, as in Iraq, no Ariadne's thread to guide them out. Jeannette Ingberman and Papo Colo line the space with 'The Labyrinth Wall: From Mythology to Reality,' inviting fifty-one artists to cover its sixty-two panels."

— "Marlene Dumas, The Labyrinth Wall, and Emily Jacir"

Haber  (ibid .) also describes artist Marlene Dumas, a recent winner of a Royal Swedish Academy Schock Prize. For a fellow Schock winner— mathematician Michael Aschbacher— see Thursday's Design.)

* For another version of the title, see this morning's front page.

Friday, August 26, 2011

Abel Prize

Filed under: Uncategorized — m759 @ 3:57 PM

The previous posts, Design and Solomon's Labyrinth,
refer, respectively,  to concepts of Tits ("buildings") and
of Thompson (imagining a future Origin of Groups ).

This suggests a review of Norway's 2008 Abel Prize,
presented to Thompson and Tits on May 20, 2008.

Poster display before the 2008 Abel Prize ceremony—

Click to enlarge.

http://www.log24.com/log/pix11B/110826-PosterDisplay.jpg

A poster of sorts in this journal on the same day, May 20, 2008—

Bright Star –

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte

– Rubén Darío  

Bright Star and Crystal Skull

Image adapted from
Blue Star Traders

Related material— Epiphany Revisited, Four Winds,
and Where Entertainment is God (continued).

Solomon’s Labyrinth

Filed under: Uncategorized — m759 @ 1:20 PM

Some context for last night's post on group theorist Michael Aschbacher—

IMAGE- Ronald Solomon on the labyrinthine classification of finite simple groups

Thursday, August 25, 2011

Design

Filed under: Uncategorized — Tags: — m759 @ 11:07 PM

"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)

Today's American Mathematical Society home page—

IMAGE- AMS News Aug. 25, 2011- Aschbacher to receive Schock prize

Some related material—

IMAGE- Aschbacher on the 2-local geometry of M24

IMAGE- Paragraph from Peter Rowley on M24 2-local geometry

The above Rowley paragraph in context (click to enlarge)—

IMAGE- Peter Rowley, 2009, 'The Chamber Graph of the M24 Maximal 2-Local Geometry,' pp. 120-121

"We employ Curtis's MOG
 both as our main descriptive device and
 also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122

And the MOG incorporates the
Geometry of the 4×4 Square.

For this geometry's relation to "design"
in the graphic-arts sense, see
Block Designs in Art and Mathematics.

Wednesday, August 24, 2011

Symmetry

Filed under: Uncategorized — m759 @ 11:07 PM

An article from cnet.com tonight —

For Jobs, design is about more than aesthetics

By: Jay Greene  

… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.

Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod,  Jobs laid out his vision for product design.

''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''

Related material: Open, Sesame Street  (Aug. 19) continues… Brought to you by the number 24

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

Conjure

Filed under: Uncategorized — Tags: — m759 @ 2:02 PM

In the catacomb of my mind
Where the dead endure—a kingdom
I conjure by love to rise

Samuel Menashe, as quoted by
Stephen Spender in a review of four
different poets, "The Last Ditch,"
The New York Review of Books , July 22, 1971

"…the ghost reveals that the beggar
is in fact a sorcerer, a necromancer
who is preparing the mandala in order
to achieve an evil end. The ascetic
intends to bind the ghost to the corpse,
place it in the center of the circle,
and worship it as a deity."

The King and the Corpse  (from synopsis in
"How Many Facets Can a Non-Existent Jewel Have?")

Menashe died on Monday, August 22, 2011.

Related material by and for two other poets
who also died on Monday:

  1. By Jerry Leiber— "Love Potion #9"
  2. For Nick AshfordNicole Kidman in
    Sermon (from Jan. 9) and
    Conjure Wife, a 1943 tale by Fritz  Leiber

See also an excerpt from Kerouac I cached on Monday, and

Men ask the way to Cold Mountain
Cold Mountain: there's no through trail .

Tuesday, August 23, 2011

Four Winds

Filed under: Uncategorized — m759 @ 11:07 AM

A Quilt Version

IMAGE- Four Winds quilt block

A Mathematical Version

IMAGE- The eight Galois quaternions

Related remarks —

For the eight-limbed star at the top of the quaternion array above,
see "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun, a slim
sceptre, and a delicate branding iron—especially when its
tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth
dimension or only die in it.”

— Fritz Leiber, short story, 1959

See also Feb. 19, 2011.

Sunday, August 21, 2011

Cuatro Vientos

Filed under: Uncategorized — m759 @ 1:26 PM

Video: Pope to youths in Cuatro Vientos: "Thank you for your joy and resistance" YouTube

Tony Long at Wired.com, Aug. 30, 2007—  "Scattered to the four winds"—

Click to enlarge:

http://www.log24.com/log/pix11B/110821-FourWindsKerouac-500w.jpg

"This Jack, joke, poor potsherd, ' patch, matchwood, immortal diamond,
                Is immortal diamond." —Gerard Manley Hopkins, Society of Jesus

Relativistic Truth

Filed under: Uncategorized — m759 @ 12:25 PM

Today's online New York Times on the conclusion of the Roman Catholic Church's "World Youth" week—

"At the end of Sunday’s Mass, the pope announced that the next such event would be in Rio de Janeiro in 2013. Until then, he told those at the service, in Portuguese, that they 'will be swimming against the tide in a society with a relativistic culture, which wishes neither to seek nor hold on to the truth.'*"

* Fact check— This agrees with the Vatican Radio version.

Related material: Relativity Blues and Portal to 1937

IMAGE- Hotel Bella Vista as 'Portal del Aguila de Oro'

The "Portal" link above is in honor of the May 2 dies natalis of Salomon Bochner (pdf).
For some background, see yesterday's Castles in the Air and Bochner in this journal.

Saturday, August 20, 2011

Castle Rock

Filed under: Uncategorized — m759 @ 6:29 PM

Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)

"The metaphor for metamorphosis…" —Endgame

Related material:

"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."

Scholarly paper on "Correlative Cosmologies"

"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.

The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….

'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."

— Alastair Macaulay in today's New York Times

SOCRATES: Let us turn off the road a little….

Libretto for Mark Morris's 'Socrates'

See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:

A 2x4 array of squares

The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S8
acting on the above eight-part figure.

See also The Moore Correspondence from last year
on today's date, August 20.

For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—

http://www.log24.com/log/pix11B/110820-Parallelisms60.jpg

"At the still point, there the dance is."
               — Four Quartets

Castles in the Air

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace

"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau

"King Solomon's Mines," 1937—

Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

The image above is an illustration from  "Romancing the Hyperspace," May 4, 2010.

Happy birthday to the late Salomon Bochner.

Problem

Filed under: Uncategorized — m759 @ 10:00 AM

Wednesday's Marginal Remarks pictured Robert De Niro
and Sean Penn in "We're No Angels." De Niro appeared
again in a Saturday Night Live sketch linked to
in last night's 9:29 post.

Here are some remarks featuring Penn related to
Peter J. Cameron's description yesterday of Sudoku
as an example of mathematics.

(Recall that the symbol #, known as 'hash,"
can stand for checkmate.)

"Chess problems are the hymn-tunes of mathematics."
— G.H. Hardy, A Mathematician's Apology

http://www.log24.com/log/pix11A/110611-TickTickHash.gif

IMAGE-Sean Penn with Nicole Kidman in 'The Interpreter'
Click to enlarge.

My card.

For a sample chess problem, see a post from Oct. 10, 2005,
the day that the Sudoku remark Cameron describes was
in the news.

Friday, August 19, 2011

Convoluted Narrative

Filed under: Uncategorized — Tags: — m759 @ 9:29 PM

IMAGE- Chilean filmmaker Raul Ruiz died Friday, Aug. 19, 2011.  Known for 'convoluted narratives.'

Related material:  Duende meets Saturday Night Live

IMAGE- Imaginary novel, 'The Pinochet Sudoku,' from Saturday Night Live De Niro sketch

The "duende" link above leads to a post containing the following—

IMAGE- Excerpt from Dec. 11, 2006, post on Pinochet and the Escorial- where Lorca said 'geometry abuts with a dream.'

For the Sudoku part, see this afternoon's Geezer Puzzle and a comment
at Diamond Geezer's weblog this morning by combinatorialist Peter J. Cameron—

This reminds me of an incident a few years ago when Sir Michael Atiyah was interviewed by a journalist, who asked him what he thought of the Sudoku craze. Sir Michael replied that he was delighted to see so many people doing mathematics every day, and was taken to task by the journalist because "there is no mathematics in it: you don't add the numbers or anything".

 

Anyway, I consider this a mathematical puzzle; I even have some fancy words for it (a Graeco-Latin square with two disjoint diagonals and some entries prescribed). But don't let that scare anyone off trying the puzzle!

Thanks, DG: I put a link to it right away.

See also the Pope's schedule today.

Geezer Puzzle

Filed under: Uncategorized — Tags: — m759 @ 2:02 PM

An RSS item today—

Peter Cameron Diamond squares Fri Aug 19, 2011 05:36 [EDT] from Peter Cameron by Peter Cameron

If you like Latin squares and such things, take a look at Diamond Geezer’s post for today: a pair of orthogonal Latin squares with two disjoint common transversals, and some entries given (if you do the harder puzzle).

 

The post referred to—

http://www.log24.com/log/pix11B/110819-DiamondGeezerPuzzle.jpg

"This Jack, joke, poor potsherd, ' patch, matchwood, immortal diamond,
                Is immortal diamond." —Gerard Manley Hopkins, Society of Jesus

Those now celebrating the Catholic Church's "World Youth" week in Madrid
may prefer a related puzzle for younger and nimbler minds:

The Diamond 16 Puzzle.

Open, Sesame Street

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Y:

"When you come to a fork in the road, take it."  —Yogi Berra

Related material:  Alice in the Garden and The Pythagorean Letter.

Z:

Zorro in this journal.

For some other symbology, see selected posts from today's date, Aug. 19.

Crux

Filed under: Uncategorized — Tags: — m759 @ 11:30 AM

"Francis Bacon used the phrase instantia crucis,  'crucial instance,' to refer to something in an experiment that proves one of two hypotheses and disproves the other. Bacon's phrase was based on a sense of the Latin word crux,  'cross,' which had come to mean 'a guidepost that gives directions at a place where one road becomes two,' and hence was suitable for Bacon's metaphor."

The American Heritage® Dictionary of the English Language, Fourth Edition

Such a cross: St. Andrew's.  Some context—

http://www.log24.com/log/pix11B/110819-XMarksTheSpot.JPG

X Marks the Spot  scene, "The Last Crusade"

Related symbology for Dan Brown—

Neville's Labrys and Notes on Mathematics and Narrative.

Thursday, August 18, 2011

Banderas*

Filed under: Uncategorized — m759 @ 8:30 PM

For Ms. Julie and the Pope

For Ms. Julie:

Nuevas Banderas in this journal…

Click on image
for details.

See also Balakrishnan's Last Problem

http://www.log24.com/log/pix11B/110818-Balakrishnan-Banners-500w.jpg

For the Pope:

http://www.log24.com/log/pix11B/110818-Legend.jpg

Now playing; click poster for details.

See also "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun, a slim
sceptre, and a delicate branding iron—especially when its
tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth
dimension or only die in it.”

— Fritz Leiber, short story, 1959

* For a time-leap from Leiber's 1959 to Hollywood's 2011, see yesterday's
  Marginal Remarks,  "The God particle ?" and a different Banderas.

Wednesday, August 17, 2011

Marginal Remarks

Filed under: Uncategorized — m759 @ 11:17 AM

Today's Google Doodle is in honor of Fermat's birthday—

http://www.log24.com/log/pix11B/110817-GoogleFermatDoodle.jpg

"I have discovered a truly marvelous proof of this theorem,
 which this doodle is too small to contain."
— Google's caption

Another marginal remark, from a link target in last night's "Ein Kampf"—

"We are talking about the spatial and temporal phenomenon of language,
not about some non-spatial, non-temporal chimera [Note in margin:
Only it is possible to be interested in a phenomenon in a variety of ways]."

— Ludwig Wittgenstein, Philosophical Investigations  (1953),  Section 108

Related material on spatial and temporal phenomena—

A Dec. 29, 2010, comment to a Dec. 26 weblog post on
"Unexpected Connections in Mathematics"—

http://www.log24.com/log/pix11B/110817-FourColor.jpg

Connoisseurs of synchronicities  in the phenomena of language may note that
these December dates mark the feasts of St. Stephen and St. Thomas Becket.

From the feast of the latter, two Log24 posts: Toy Stories and True Grid.

Those less enchanted by pop math than Google may prefer to observe
two other birthdays today— those of Robert De Niro and of Sean Penn:

http://www.log24.com/log/pix11B/110817-NoAngels.jpg

Ein Kampf

Filed under: Uncategorized — m759 @ 1:00 AM

"Philosophy is a battle against the bewitchment of our intelligence by means of our language."

"Die Philosophie ist ein Kampf gegen die Verhexung unsres Verstandes durch die Mittel unserer Sprache."

— Ludwig Wittgenstein, Philosophical Investigations  (1953),  Section 109

"When the battle's lost and won" — The Scottish play

Related material— Monday's Language Game.

Tuesday, August 16, 2011

Harrowing (continued)

Filed under: Uncategorized — m759 @ 12:00 PM

Theology in 'A Flag for Sunrise'

There is an unwarranted leap here
from "suggests" to "knowledge."

See Under the Volcano  and "harrowing" in this journal.

Vets Club

Filed under: Uncategorized — m759 @ 1:00 AM

http://www.log24.com/log/pix11B/110816-NYTobits.jpg

"Let him who is without sin among you
 serve the first drink."

Monday, August 15, 2011

Language Game

Filed under: Uncategorized — m759 @ 3:33 PM

http://www.log24.com/log/pix11B/110815-OmegaPGA.jpg

Related material—

Yesterday, 3:33 PM, in this journal— "Time for you to see the field"— and…

Faulkner's The Sound and the Fury , opening paragraph of part two, "June Second, 1910"—

"When the shadow of the sash appeared on the curtains it was between seven and eight oclock and then I was in time again, hearing the watch. It was Grandfather's and when Father gave it to me he said I give you the mausoleum of all hope and desire; it's rather excruciating-ly apt that you will use it to gain the reducto absurdum of all human experience which can fit your individual needs no better than it fitted his or his father's. I give it to you not that you may remember time, but that you might forget it now and then for a moment and not spend all your breath trying to conquer it. Because no battle is ever won he said. They are not even fought. The field only reveals to man his own folly, and despair, and victory is an illusion of philosophers and fools."

See also Willard Van Orman Quine in this journal on August 15, 2009

"A tale told by an idiot"— and such a tale—

http://www.log24.com/log/pix11B/110815-FaulknerApril7.jpg

Sunday, August 14, 2011

Sunday Review

Filed under: Uncategorized — Tags: — m759 @ 3:33 PM

The Sunday New York Times  today—

http://www.log24.com/log/pix11B/110814-GablerNYT500w.jpg

This suggests…

The Elusive Small Idea—

Part I:

McLuhan and the Seven Snow Whites

http://www.log24.com/log/pix11B/110814-GablerNYT500w7white.jpg

Part II (from "Marshall, Meet Bagger," July 29):

"Time for you to see the field."

http://www.log24.com/log/pix11B/110814-TheFieldGF8.jpg

For further details, see the 1985 note
"Generating the Octad Generator."

McLuhan was a Toronto Catholic philosopher.
For related views of a Montreal Catholic philosopher,
see the Saturday evening post.

Saturday, August 13, 2011

Taylor Made

Filed under: Uncategorized — m759 @ 7:11 PM

The Montreal Catholic philosopher Charles Taylor has been reviewed
online recently in The New Yorker  and The Nation .

Here is some background from this journal and from a 1989 book.

Update of 5:01 AM August 14— See also Valéry and the Self in this journal.

Friday, August 12, 2011

Into the Woods

Filed under: Uncategorized — Tags: — m759 @ 8:48 AM

"What’s best about us, I hope, is that we teach them
the ‘forest of symbols,’ to borrow deliberately from
a poem called ‘Correspondences,’ by Baudelaire."

The late Stanley Bosworth, founding headmaster
    of St. Ann's School in Brooklyn

Bosworth died Sunday.

Related material—

  1. Saturday's Correspondences,
  2. Sunday's   Coordinated Steps, and
  3. Monday's  Organizing the Mine Workers.

Wednesday, August 10, 2011

Objectivity

Filed under: Uncategorized — m759 @ 12:25 PM

From math16.com

Quotations on Realism
and the Problem of Universals:

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268

"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.

"People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only 'truths' strictly worthy of the name. Such truths I will call 'diamonds'; they are highly desirable but hard to find….The happy metaphor is Morris Kline's in Mathematics in Western Culture (Oxford, 1953), p. 430."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 114 and 117

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call 'models of reality'). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 256 and 259

Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is non-Euclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000

The correct statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)— 

From The Search for Invariants (June 20, 2011):

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

http://www.log24.com/log/pix11B/110810-MajaLovrenovBio.jpg

Related material from Sunday's New York Times  travel section—

"Exhibit A is certainly Ljubljana…."

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: Uncategorized — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Organizing the Mine Workers

Filed under: Uncategorized — m759 @ 12:24 PM

From this journal on Saturday, August 6, 2011

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances " (in The Flowers of Evil )

From the New York Times  philosophy column "The Stone" earlier that day

"… a magnificent and colorful parade of disorganized and rhapsodic thoughts"

— Baudelaire

From Uncle Walt— (See yesterday's "Coordinated Steps")—

http://www.log24.com/log/pix11B/110808-DwarfsParade500w.jpg

For a better organized, less rhapsodic parade, see Saturday's Correspondences.

Sunday, August 7, 2011

Coordinated Steps

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

From this morning's Sunday New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From this evening's news—

"TOKYO (Dow Jones)–The Group of Seven leading industrial nations said they are committed to taking coordinated steps…."

Who's their choreographer?

My choice would be Uncle Walt—

'The Seven Dwarfs and their Diamond Mine

See also yesterday's "Norway Summer" and the link in
this afternoon's "Reflection Group" to "A Better Story."

Reflection Group

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

The European Reflection Group report of Saturday, May 8, 2010—
"Project Europe 2030: Challenges and Opportunities" (pdf, 46 pp.)—

"All our members agree on one fundamental issue:
Europe is currently at a turning point in its history."

This journal on the same date— "A Better Story"—

"…I can imagine the decisive evolutionary beginnings of humans and societies… not in an adult version, but in the playful mentality of children…. An unlikely story? Perhaps. I am looking out for a better story."

Hans G. Furth, Desire for Society: Children's Knowledge as Social Imagination, published by Springer, 1996, p. 181

"The clocks were striking thirteen." — George Orwell

See also this journal during the rest of May 2010 and "Sermon" from Sunday, February 20, 2011.

Saturday, August 6, 2011

Happy Web Day

Filed under: Uncategorized — m759 @ 9:00 PM

Today the World Wide Web turns 20.

http://www.log24.com/log/pix11B/110806-EightfoldGeometrySearch.jpg

See also Galois Memorial and Correspondences.

Norway Summer

Filed under: Uncategorized — m759 @ 4:00 PM

(Continued from June 21)

Footnote to a new web page from the European Culture Congress—

Photo credit: Josefine Lyche, “The 2×2 Case (Diamond Theorem)
after Steven H. Cullinane”, 450 x 650 cm,
Tromsø Kunstforening, 2010, image courtesy: the artist.

Correspondences

Filed under: Uncategorized — Tags: , — m759 @ 2:00 PM

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.

 

For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

Thursday, August 4, 2011

Midnight in Oslo

Filed under: Uncategorized — m759 @ 6:00 PM

For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth

(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)

http://www.log24.com/log/pix11B/110804-Pesic-PlatosDiamond.jpg

Plato's Diamond

The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—

Paradigms

"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate….

… Both in the Republic  and in the Sophist  there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno  that the square on the diagonal is double the original square in area."

– William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985

Plato's paradigm in the Meno

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Aspects of the paradigm change—

Monochrome figures to
   colored figures

Areas to
   transformations

Continuous transformations to
   non-continuous transformations

Euclidean geometry to
   finite geometry

Euclidean quantities to
   finite fields

The 24 patterns resulting from the paradigm change—

http://www.log24.com/log/pix11B/110805-The24.jpg

Each pattern has some ordinary or color-interchange symmetry.

This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem.

Related material: Plato's Diamond by Oslo artist Josefine Lyche.

Plato’s Ghost  evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”

— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics  (by Jeremy Gray, September 2008)

"Remember me to her."

— Closing words of the Algis Budrys novel Rogue Moon .

Background— Some posts in this journal related to Abel or to random thoughts from his birthday.

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