Log24

Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: General,Geometry — Tags: — m759 @ 12:35 PM

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .

DOI

Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM
 

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by  V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Friday, August 1, 2014

The Diamond-Theorem Correlation

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

Click image for a larger, clearer version.

IMAGE- The symplectic correlation underlying Rosenhain and Göpel tetrads

Tuesday, December 3, 2013

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A new website illustrates its URL.
See DiamondSpace.net.

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

Monday, February 11, 2013

The Penrose Diamond

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

IMAGE- The Penrose Diamond

Related material:

(Click to enlarge.)

See also remarks on Penrose linked to in Sacerdotal Jargon.

(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)

Thursday, May 31, 2012

Black Diamond

Filed under: General — Tags: — m759 @ 12:26 PM

IMAGE- Four-elements-diamond test problem in the style of Raven's Progressive Matrices (answer: the black diamond)

“To say more is to say less.”
― Harlan Ellison, as quoted at goodreads.com

Saying less—

Saturday, March 17, 2012

The Purloined Diamond

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

(Continued)

The diamond from the Chi-rho page
of the Book of Kells —

The diamond at the center of Euclid's
Proposition I, according to James Joyce
(i.e., the Diamond in the Mandorla) —

Geometry lesson: the vesica piscis in Finnegans Wake

The Diamond in the Football

Football-mandorla

“He pointed at the football
  on his desk. ‘There it is.’”
         – Glory Road
   

Wednesday, December 21, 2011

The Purloined Diamond

Filed under: General — Tags: , — m759 @ 9:48 AM

Stephen Rachman on "The Purloined Letter"

"Poe’s tale established the modern paradigm (which, as it happens, Dashiell Hammett and John Huston followed) of the hermetically sealed fiction of cross and double-cross in which spirited antagonists pursue a prized artifact of dubious or uncertain value."

For one such artifact, the diamond rhombus formed by two equilateral triangles, see Osserman in this journal.

Some background on the artifact is given by John T. Irwin's essay "Mysteries We Reread…" reprinted in Detecting Texts: The Metaphysical Detective Story from Poe to Postmodernism .

Related material—

Mathematics vulgarizer Robert Osserman died on St. Andrew's Day, 2011.

A Rhetorical Question

Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer

http://www.log24.com/log/pix11C/111130-SunshineCleaning.jpg

Above: Amy Adams in "Sunshine Cleaning"

Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest  thing there is
(Tractatus Logico-Philosophicus  No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach  sein. Sie ist vor  aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh.  No. 5.5563.)

See also

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

* Updates of 9:29 PM ET July 10, 2011—

A  mnemonic  from a course titled "Galois Connections and Modal Logics"—

"Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and diamond (diamond) for possibly. Then, for example, diamondP  can be read as
'it is possibly the case that P .'"

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

"The diamond symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’"

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

Monday, June 11, 2018

Glitter

Filed under: General,Geometry — Tags: — m759 @ 8:32 PM

A Scientific American  headline today —

Glittering Diamond Dust in Space
Might Solve a 20-Year-Old Mystery

Related art —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Background:  "Diamond Dust" + Glitter in this journal.

Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Friday, December 8, 2017

Logos (Continued)

Filed under: General — Tags: — m759 @ 3:00 PM

Nietzsche, 'law in becoming' and 'play in necessity'

"Denn die Welt braucht ewig die Wahrheit,
also braucht sie ewig Heraklit:
obschon er ihrer nicht bedarf.
Was geht ihn sein Ruhm an?
Der Ruhm bei »immer fortfließenden Sterblichen!«,
wie er höhnisch ausruft.
Sein Ruhm geht die Menschen etwas an, nicht ihn,
die Unsterblichkeit der Menschheit braucht ihn,
nicht er die Unsterblichkeit des Menschen Heraklit.
Das, was er schaute, die Lehre vom Gesetz im Werden
und vom
Spiel in der Notwendigkeit 
, muß von jetzt
ab ewig geschaut werden: er hat von diesem größten
Schauspiel den Vorhang aufgezogen."

Logos for Philosophers
(Suggested by Modal Logic) —

Nietzsche, 'law in becoming' and 'play in necessity'

Saturday, October 28, 2017

Lowell Brown at Vanity Fair

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:18 PM

A sequel to the post  CP  is for Consolation Prize  (Sept. 3, 2016)

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry

Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: , — m759 @ 12:31 PM

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices

Related material from a post of Aug. 6, 2014

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

(Here "five point sets" should be "five-point sets.")

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate 
abuse of language .  For the real definition of 
“symplectic structure,” see (for instance) 
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry 
, Vol 2.) To establish that the above
figure is indeed symplectic , see the post 
Zero System of July 31, 2014.

Saturday, June 4, 2016

Icons

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

From this morning's news, a  cultural icon —

From November 18, 2015, four  icons —

— the three favicons above, and the following:

Jack in the Box, by Natasha Wescoat

Sunday, May 15, 2016

One Ring

Filed under: General,Geometry — Tags: , — m759 @ 5:06 PM

(Continued from May 11 and May 15.)

Poem by Eleanor Wilner from 'Reversing the Spell' speaks of diamonds and 'glitter.' (Pbk. publ. Nov. 1, 1997)

Friday, April 29, 2016

Blackboard Jungle…

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Continues .

An older and wiser James Spader —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Saturday, January 23, 2016

Hard

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

"Hard Science Fiction in the era of short attention spans,
crowd-sourcing, and rapid obsolescence"

— May 26, 2012, Dragon Press Bookstore symposium

Related material:  Posts now tagged Black Diamond.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Friday, January 22, 2016

Story

Filed under: General — Tags: — m759 @ 10:21 AM

The New Yorker , April 12, 2004 —

Wednesday, November 18, 2015

Tuesday, November 17, 2015

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

Filed under: General,Geometry — Tags: , — m759 @ 2:45 AM



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: General,Geometry — Tags: — m759 @ 2:01 AM

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

 

Saturday, July 4, 2015

Context

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks 
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .

See also the diamond-theorem correlation.  

Saturday, June 13, 2015

Egg Tales

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

"And not all the king's men nor his horses
 Will resurrect his corpus."

Finnegans Wake

See as well Andy Weir's "The Egg" and Working Backward.

Wednesday, February 25, 2015

Words and Images

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

The words:  "symplectic polarity"—

The images:

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

Picturing the Smallest Projective 3-Space

Quilt Block Designs

Saturday, February 21, 2015

High and Low Concepts

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

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European. 
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

Sunday, August 24, 2014

Symplectic Structure…

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the
3-subsets.

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Wednesday, August 13, 2014

Symplectic Structure continued

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —

Related material in this journal (click image for posts) —

Monday, August 11, 2014

Syntactic/Symplectic

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM

(Continued from August 9, 2014.)

Syntactic:

Symplectic:

"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive 
. They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."

– Susanne K. LangerPhilosophy in a New Key

For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).

This is a symplectic  correlation,* constructed using the following
visual structure:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.

Wednesday, August 6, 2014

Symplectic Structure*

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens  54, 59-79 (1992):

"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: , — m759 @ 6:11 PM

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Sunday, June 8, 2014

Vide

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Friday, February 21, 2014

Night’s Hymn of the Rock

Filed under: General,Geometry — Tags: , , — m759 @ 3:33 AM

One way of interpreting the symbol  IMAGE- Modal Diamond in a square 
at the end of yesterday's post is via
the phrase "necessary possibility."

See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .

The Tablet  post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—

IMAGE- Tom Wolfe in 'The Painted Word' on conceptual art

On that  day (July 23) another weblog had
a post titled

Wallace Stevens: Night's Hymn of the Rock.

Some related narrative —

IMAGE- The 2001 film 'The Discovery of Heaven'

I prefer the following narrative —

Part I:  Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"

Part II:  Mystery Box III: Inside, Outside (2014)

Thursday, February 20, 2014

Relativity Blues

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

(Continued

A review of this date in 2005 —

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

— Keith Allen Korcz

And what do we  
   symbolize by   The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

Wednesday, December 18, 2013

Bing Bang Theory

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

Microsoft in 2009 on its new search engine name—

"We like Bing because it sounds off in our heads
when we think about that moment of discovery
and decision making— when you resolve those
important tasks."

A search on Bing today —

IMAGE- Top search result on Bing for 'diamond space' on Dec. 18, 2013

A colorful tale —

IMAGE- The Diamond 16 Puzzle, with commentary

"Bing bang, I saw the whole gang
Bobby Darin, 1958

Thursday, December 12, 2013

Outsider Art

Filed under: General,Geometry — Tags: — m759 @ 4:10 PM

"… Galois was a mathematical outsider…."

— Tony Mann, "head of the department of mathematical sciences,
University of Greenwich, and president, British Society for the
History of Mathematics," in a May 6, 2010, review of Duel at Dawn
in Times Higher Education.

Related art: 

(Click for a larger image.)

IMAGE- Google search for 'Diamond Space' + Galois

For a less outside  version of the central image
above, see Kunstkritikk  on Oct. 15, 2013.

Monday, August 12, 2013

Form

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest 
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Saturday, June 29, 2013

Thursday, June 27, 2013

Tuesday, May 28, 2013

Codes

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Transgressing the Boundary

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

The title refers not to the 1996 Sokal hoax (which has
Boundaries , plural, in the title), but to the boundary
discussed in Monday's Penrose diamond post

"Science is a differential equation.
Religion is a boundary condition."

Alan Turing in the epigraph to the
first chapter of a book by Terence Tao

From the Tao book, page 170—

"Typically the transformed solution extends to the
boundary of the Penrose diamond and beyond…."

Transgressing the boundary between science
and religion is the topic of a 1991 paper available
at JSTOR for $29.

For the Pope on Ash Wednesday:

"Think you might have access 
to this content via your library?" —JSTOR

See also Durkheim at Harvard.

Friday, February 1, 2013

Get Quotes

Filed under: General — Tags: , — m759 @ 4:01 PM

For Tony Kushner fans:

For logic fans:

IMAGE- NY Times market quotes, American Express Gold Card ad, Kevin Spacey in 'House of Cards' ad

John Searle on Derrida:

On necessity, possibility, and 'necessary possibility'

In the box-diamond notation, the axiom Searle quotes is

.

"The euclidean property guarantees the truth of this." — Wikipedia

Linking to Euclid

Clicking on "euclidean" above yields another Wikipedia article

"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."

Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.

Tuesday, January 22, 2013

Raven Light

Filed under: General — Tags: , — m759 @ 11:40 AM

"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of things. …

…matrices are considered the gold standard of fluid-intelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."

— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012

See also "Raven Steals the Light" in this  journal.

Related material:

Plan 9 from MIT and, perhaps exemplifying crystallized  rather than fluid  intelligence, Black Diamond.

Thursday, November 1, 2012

For All Saints’ Day

Filed under: General — Tags: , — m759 @ 5:31 AM

Conclusion of "The Storyteller," a story 
by Cynthia Zarin about author Madeleine L'Engle—

The New Yorker , April 12, 2004 —

Note the black diamond at the story's end.

Monday, September 17, 2012

The Count

Filed under: General — Tags: , — m759 @ 11:01 PM
 

… I saw a shadow
sliding around the ropes
to get at me. The referee
moved it back, and then
went over and picked up the count.
"One!" The fog was clearing.

I rose to a knee,
and at "nine" to my feet.

— Louis Simpson, "The Appointment"

Simpson reportedly died on Holy Cross Day.

That day in this journal—

IMAGE- Log24 posts 'Please Mister Please' and 'Plan 9'

Saturday, May 26, 2012

Talk Amongst Yourselves

Filed under: General,Geometry — Tags: , — m759 @ 3:33 PM

Hard Science Fiction weekend at Dragon Press Bookstore

Saturday May 26:
11am-noon Playing with the net up:
Hard Science Fiction in the era of
short attention spans, crowd-sourcing,
and rapid obsolescence
( Greg Benford, James Cambias, Kathryn Cramer)
….
3pm-4:30 Technological optimism and pessimism;
utopia and dystopia; happy endings & sad endings:
what do these oppositions have to do with one another?
Are they all the same thing? How are they different
from one another? Group discussion.

My own interests in this area include…

(Click image for some context)

IMAGE- 'The Stars My Destination' (with cover slightly changed)

    The above was adapted from a 1996 cover

IMAGE- PyrE on the 1996 Vintage Books cover of 'The Stars My Destination'

 Vintage Books, July 1996. Cover: Evan Gaffney.

For the significance of the flames, 
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.

Sunday, April 1, 2012

The Palpatine Dimension

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

http://www.log24.com/log/pix10B/101128-TheEmbedding.gif

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

See also uses of the word triality  in mathematics. For instance…

A discussion of triality by Edward Witten

Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6  thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron

There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality  link from above
and Cameron's Klein Correspondence and Triality.)

Cameron applies his  triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Saturday, October 15, 2011

Synergy

Filed under: General — m759 @ 6:48 AM

"Examples are the stained-glass windows of knowledge." —Nabokov

Suggested by yesterday's evening NY lottery

Post 4248: The Hunt for Exemplary October, and
Post   942: Links for St. Benedict

Related material—

three-point landing n

1. (Engineering / Aeronautics) an aircraft landing
in which the two main wheels and the nose or tail wheel
all touch the ground simultaneously

— Collins English Dictionary

See also…

     Tiffany Case and…

 The Diamond
in the Mandorla

Football-mandorla with link to 'Heaven Can<br />
Wait'

“He pointed at the football
  on his desk. ‘There it is.’”
     – Glory Road
    

Friday, September 24, 2010

Heisman Trophy

Filed under: General — m759 @ 6:00 PM

An Ecumenical Hymn

For those who observed Yom Kippur at
Harvard's Memorial Church on Saturday,
September 18, 2010—

Friday night and the lights are low…

http://www.log24.com/log/pix10B/100924-NYsunset.jpg

NASA Astronomy Picture of the Day for
July 13, 2007— Manhattanhenge.

See also on July 13, 2007, in this  journal, a post
for Harrison Ford's 65th birthday featuring the
ecumenical diamond-in-a-football religious symbol—

The image “http://www.log24.com/log/pix07/070713-Ford2.jpg” cannot be displayed, because it contains errors.
 
Related story: Harvard Defeats Holy Cross 34 – 6.

Wednesday, July 14, 2010

Happy Bastille Day…

Filed under: General — m759 @ 11:00 AM

To the leftist philosophers of Villanova

From "Make a Différance"
(Women's History Month, 2005)—

Frida Saal's 

Lacan The image 
“http://www.log24.com/log/pix05/050322-Diamond.gif” cannot be displayed,
 because it contains errors. Derrida:

"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names….  What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."

Football-mandorla (vesica piscis) with link to 'Heaven Can 
Wait'

“He pointed at the football
  on his desk. ‘There it is.’”
Glory Road
    

Quodlibet* 

Compare and contrast
the diamond in the football
with the jewel in the lotus.

* "A scholastic argumentation upon a subject chosen at will, but almost always theological. These are generally the most elaborate and subtle of the works of the scholastic doctors." —Century Dictionary

Saturday, March 7, 2009

Saturday March 7, 2009

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

One or Two Ideas
 
Today's birthday: Piet Mondrian
 
From James Joyce's A Portrait of the Artist as a Young Man:

he hearth and began to stroke his chin.

–When may we expect to have something from you on the esthetic question? he asked.

–From me! said Stephen in astonishment. I stumble on an idea once a fortnight if I am lucky.

–These questions are very profound, Mr Dedalus, said the dean. It is like looking down from the cliffs of Moher into the depths. Many go down into the depths and never come up. Only the trained diver can go down into those depths and explore them and come to the surface again.

–If you mean speculation, sir, said Stephen, I also am sure that there is no such thing as free thinking inasmuch as all thinking must be bound by its own laws.

–Ha!

–For my purpose I can work on at present by the light of one or two ideas of Aristotle and Aquinas.

–I see. I quite see your point.

Besides being Mondrian's birthday, today is also the dies natalis (in the birth-into-heaven sense) of St. Thomas Aquinas and, for those who believe worthy pre-Christians also enter heaven, possibly of Aristotle.

Pope Benedict XVI explained the dies natalis concept on Dec. 26, 2006:

"For believers the day of death, and even more the day of martyrdom, is not the end of all; rather, it is the 'transit' towards immortal life. It is the day of definitive birth, in Latin, dies natalis."

The Pope's remarks on that date
were in St. Peter's Square.
 
From this journal on that date,
a different square —
 
The Seventh Symbol:
 

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)

The square may be regarded as
symbolizing art itself.
(See Nov.30 – Dec.1, 2008.)

In honor of
Aristotle and Aquinas,
here is a new web site,
illuminati-diamond.com,
with versions of the diamond shape
made famous by Mondrian

Cover of  Mondrian: The Diamond Compositions

— a shape symbolizing
possibility within modal logic
 as well as the potentiality of
 Aristotle's prima materia.

Monday, December 29, 2008

Monday December 29, 2008

Filed under: General — Tags: — m759 @ 12:21 PM
The Gift
 

Plato's Diamond

Robert Stone,
A Flag for Sunrise:

"'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.'

Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion."

For further details, click on the diamond.

 

Related narratives:

Today's online Times on
the Saturday, Dec. 27,
death of an artist:

Robert Graham obituary, NY Times, 12/29/08

"Dale Wasserman… the playwright responsible for two Broadway hits of the 1960s, 'One Flew Over the Cuckoo’s Nest' and 'Man of La Mancha,' died on Sunday [December 21, 2008] at his home in Paradise Valley, Ariz., near Phoenix….

Mr. Wasserman wrote more than 75 scripts for television, the stage and the movies, including screenplays for 'The Vikings' (1958), a seafaring epic with Tony Curtis and Kirk Douglas, and 'A Walk With Love and Death' (1969), a John Huston film set in 14th-century Europe….

He feuded with… John Huston, who gave the lead female role in 'Walk' to his teenage daughter, Anjelica, against Mr. Wasserman's wishes. And he never attended ceremonies to receive the awards he won."

Accepting for Mr. Wasserman:
Mr. Graham's widow,
Anjelica Huston

Anjelica Huston and Jack Nicholson

"Well…"

Tuesday, April 29, 2008

Tuesday April 29, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 11:09 AM
Sacerdotal Jargon
at Harvard:

Thomas Wolfe

Thomas Wolfe
(Harvard M.A., 1922)

versus

Rosalind Krauss

Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)

on

The Kernel of Eternity

"No culture has a pact with eternity."
George Steiner, interview in  
The Guardian of April 19

"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."

— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005

 

From today's online Harvard Crimson:

"… under the leadership of Faust,
Harvard students should look forward
to an ever-growing opportunity for
international experience
and artistic endeavor."

 

Wolfgang Pauli as Mephistopheles

Pauli as Mephistopheles
in a 1932 parody of
Goethe's
Faust at Niels Bohr's
institute in Copenhagen

From a recent book
on Wolfgang Pauli,
The Innermost Kernel:

Pauli's Dream Square (square plus the two diagonals)

A belated happy birthday
to the late
Felix Christian Klein
  (born on April 25) —

The Klein Group: The four elements in four colors, with black points representing the identity

Another Harvard figure quoted here on Dec. 5, 2002:

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."

— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)

From a review of Rosalind Krauss's The Optical Unconscious  (MIT Press hardcover, 1993):

Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid-19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."

— Arthur C. Danto in ArtForum, Summer 1993

Rosalind Kraus in The Optical Unconscious (MIT Press paperback, 1994):

For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude Lévi-Strauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.

For related non-sacerdotal jargon, see…
 

Wikipedia on the Klein group (denoted V, for Vierergruppe):

In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.

For radicals of another sort, see A Logocentric Meditation, A Mass for Lucero, and [update of 7 PM] Steven Erlanger in today's New York Times— "France Still Divided Over Lessons of 1968 Unrest."

For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:

April 29, 2008, Google search on 'penrose space time'

Click on the above
 image for details.

See also yesterday's
Religious Art.

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General — Tags: — m759 @ 12:00 PM
For Balanchine's Birthday

(continued from
January 9, 2003)

George Balanchine

Encyclopædia Britannica Article

born January 22
[January 9, Old Style], 1904,
St. Petersburg, Russia
died April 30, 1983, New York,
New York, U.S.

Photograph:George Balanchine.
George Balanchine.
©1983 Martha Swope

original name 
Georgy Melitonovich Balanchivadze

most influential choreographer of classical ballet in the United States in the 20th century.  His works, characterized by a cool neoclassicism, include The Nutcracker (1954) and Don Quixote (1965), both pieces choreographed for the New York City Ballet, of which he was a founder (1948), the artistic director, and the…


Balanchine,  George… (75 of 1212 words)

"What on earth is
a concrete universal?"
— Robert M. Pirsig

Review:

From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:

"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."

There is, however,
"the field of reason"–

the 3×3 grid:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Click on grid
for details.

As Rosalind Krauss
has noted, some artists
regard the grid as

"a staircase to
  the Universal."

Other artists regard
Epiphany itself as an
approach to
the Universal:

"Epiphany signals the traversal
of the finite by the infinite,
of the particular by the universal,
of the mundane by the mystical,
of time by eternity.
"

Richard Kearney, 2005,
in The New Arcadia Review

The image “http://www.log24.com/log/pix07/070109-Kearney2.jpg” cannot be displayed, because it contains errors.

Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.

"… one of the things that worried me about traditional metaphysics, at least as I imbibed it in a very Scholastic manner at University College Dublin in the seventies, is that philosophy was realism and realism was truth. What disturbed me about that was that everything was already acquired; truth was always a systematic given and it was there to be learned from Creation onwards; it was spoken by Jesus Christ and then published by St. Thomas Aquinas: the system as perfect synthesis. Hence, my philosophy grew out of a hunger for the 'possible' and it was definitely a reaction to my own philosophical formation. Yet that wasn't my only reaction. I was also reacting to what I considered to be the deep pessimism, and even at times 'nihilism' of the postmodern turn."

— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 2005-2006

For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:

 

"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

 

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity,
Christ Church College, Oxford
(the home of Lewis Carroll)

Friday, December 29, 2006

Friday December 29, 2006

Filed under: General — Tags: — m759 @ 11:01 AM
Tools
of Christ Church

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon

Cover of Thomas, by Shelley Mydans: Sword and its shadow, a cross

Click on picture for details.

Today is the feast
of St. Thomas Becket.

In his honor, a meditation
on tools and causation:

"Lewis Wolpert, an eminent developmental biologist at University College London, has just published Six Impossible Things Before Breakfast, a pleasant, though rambling, look at the biological basis of belief. While the book focuses on our ability to form causal beliefs about everyday matters (the wind moved the trees, for example), it spends considerable time on the origins of religious and moral beliefs. Wolpert defends the unusual idea that causal thinking is an adaptation required for tool-making. Religious beliefs can thus be seen as an odd extension of causal thinking about technology to more mysterious matters. Only a species that can reason causally could assert that 'this storm was sent by God because we sinned.' While Wolpert's attitude toward religion is tolerant, he's an atheist who seems to find religion more puzzling than absorbing."

Review by H. Allen Orr in
The New York Review of Books,
Vol. 54, No. 1, January 11, 2007    


"An odd extension"–

Wolpert's title is, of course,
from Lewis Carroll.

Related material:

"It's a poor sort of memory
that only works backwards."
Through the Looking-Glass

An event at the Kennedy Center
broadcast on
December 26, 2006
(St. Steven's Day):

"Conductor John Williams, a 2004 Honoree, says, 'Steven, sharing our 34-year collaboration has been a great privilege for me. It's been an inspiration to watch you dream your dreams, nurture them and make them grow. And, in the process, entertain and edify billions of people around the world. Tonight we'd like to salute you, musically, with a piece that expresses that spirit beautifully … It was written by Leonard Bernstein, a 1980 Kennedy Center Honoree who was, incidentally, the first composer to be performed in this hall.' Backed by The United States Army Chorus and The Choral Arts Society, soprano Harolyn Blackwell and tenor Gregory Turay sing the closing number for Spielberg's tribute and the gala itself. It's the finale to the opera 'Candide,' 'Make Our Garden Grow,' and Williams conducts."

CBS press release

See also the following,
from the conclusion to

"Mathematics and Narrative"

(Log24, Aug. 22, 2005):

Diamond on cover of Narrative Form, by Suzanne Keen

"At times, bullshit can
only be countered
   with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to at least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

"Il faut cultiver notre jardin."
— Voltaire

"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz 

Diamond in a square

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
 Regius Professor of Divinity,
  Christ Church College, Oxford
(the home of Lewis Carroll)

For further details,
click on the
Christ Church diamond.

Tuesday, October 10, 2006

Tuesday October 10, 2006

Filed under: General — Tags: , — m759 @ 8:00 PM
Mate in
Two Seconds

From Oct. 14 last year:

The image “http://www.log24.com/log/pix05B/051014-Tick.gif” cannot be displayed, because it contains errors.

From Oct. 13 last year
(Yom Kippur):

A Poem for Pinter
Oct. 13, 2005

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth.  The words 'necessary' and 'possible' are called modal connectives, or modalities.  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.  (For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.

That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.

For more on the former, see
Death in Two Seconds.

For more on the latter, see
The Halmos Tombstone.

4x9 black monolith

Monday, January 23, 2006

Monday January 23, 2006

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

The Case

An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:

From The Hustler, by Walter Tevis:

The only light in the room was from the lamp over the couch where she was reading.
    He looked at her face.  She was very drunk.  Her eyes were swollen, pink at the corners.  "What's the book?" he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
    She blinked up at him, smiled sleepily, and said nothing.
    "What's the book?"  His voice had an edge now.
    "Oh," she said.  "It's Kierkegaard.  Soren Kierkegaard."  She pushed her legs out straight on the couch, stretching her feet.  Her skirt fell back a few inches from her knees.  He looked away.
    "What's that?" he said.
    "Well, I don't exactly know, myself."  Her voice was soft and thick.
    He turned his face away from her again, not knowing what he was angry with.  "What does that mean, you don't know, yourself?"
    She blinked at him.  "It means, Eddie, that I don't exactly know what the book is about.  Somebody told me to read it, once, and that's what I'm doing.  Reading it."
    He looked at her, tried to grin at her– the old, meaningless, automatic grin, the grin that made everybody like him– but he could not.  "That's great," he said, and it came out with more irritation than he had intended.
    She closed the book, tucked it beside her on the couch.  "I guess this isn't your night, Eddie.  Why don't we have a drink?"
    "No."  He did not like that, did not want her being nice to him, forgiving.  Nor did he want a drink.
    Her smile, her drunk, amused smile, did not change.  "Then let's talk about something else," she said.  "What about that case you have?  What's in it?"  Her voice was not prying, only friendly.  "Pencils?"
    "That's it," he said.  "Pencils."
    She raised her eyebrows slightly.  Her voice seemed thick.  "What's in it, Eddie?"
    "Figure it out yourself."  He tossed the case on the couch.

 

Related material:

Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,

The Diamond of Possibility,

The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

the Baseball Almanac,

The image “http://www.log24.com/log/pix06/060123-BaseballLogo75.gif” cannot be displayed, because it contains errors.

and this morning's entry, "Natural Hustler."

Thursday, January 19, 2006

Thursday January 19, 2006

Filed under: General — Tags: — m759 @ 9:00 AM

Logos
 

The image “http://www.log24.com/log/pix06/060119-BibleLogoSmall.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06/060119-AlvinPlantinga2.jpg” cannot be displayed, because it contains errors.

Alvin Plantinga

The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

Philosophy
logo

The image “http://www.log24.com/log/pix06/060119-HarryPlantinga2.jpg” cannot be displayed, because it contains errors.

Harry Plantinga

The image “http://www.log24.com/log/pix06/060119-CCEL.gif” cannot be displayed, because it contains errors.

CCEL
logo

The image “http://www.log24.com/log/pix06/060119-LEngle2.jpg” cannot be displayed, because it contains errors.

Madeleine
L'Engle

The image “http://www.log24.com/log/pix06/060119-Tesseract2.jpg” cannot be displayed, because it contains errors.

Tesseract
logo

Thursday, October 13, 2005

Thursday October 13, 2005

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

A Poem for Pinter

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise  (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary  and possible  truth.  The words 'necessary' and 'possible' are called modal connectives , or modalities .  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.

(For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

Monday, August 22, 2005

Monday August 22, 2005

Filed under: General — Tags: — m759 @ 4:07 PM
The Hole

Part I: Mathematics and Narrative

The image “http://www.log24.com/log/pix05B/050822-Narr.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis on last month's conference on "mathematics and narrative"–

Doxiadis is describing how talks by two noted mathematicians were related to

    "… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "

Doxiadis says both talks had "a big hole in the middle."  

    "Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts.  [In one part] I’ll tell you a few things about proofs.  [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
  "And the hole?"
  "This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’  But the similarity was not clearly explained: the hole, you see, the ‘miracle.’  Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
 

Part II: Possible Worlds

"At times, bullshit can only be countered with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

 

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole
 

Part III: Modal Theology

  "'What is this Stone?' Chloe asked….
  '…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

  — Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)


"The lapis was thought of as a unity and therefore often stands for the prima materia in general."

  — Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)

"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."

 
  — The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)
 
"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:01 PM

Make a Différance

From Frida Saal's
Lacan The image “http://www.log24.com/log/pix05/050322-Diamond.gif” cannot be displayed, because it contains errors. Derrida:

"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names. For that reason their respective works become totally non-superposable and at the same time they were built with an awareness, or at least a partial awareness, of each other. What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."

 


From a Contemporary Literary Theory website:

"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."

From a text purchased on
Make a Difference Day, Oct. 23, 1999:

The image “http://www.log24.com/log/pix05/050322-Fig39.gif” cannot be displayed, because it contains errors.22. Without using the Pythagorean Theorem prove that the hypotenuse of  an isosceles right triangle will have the length The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.  if the equal legs have the length 1.  Suggestion: Consider the similar triangles in Fig. 39.
23.  The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas.  We cannot do so now because we have not yet considered the idea of area.  Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors. .

 

— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941)



Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics*  — they may in other respects be quite superior to men:

The image “http://www.log24.com/log/pix05/050322-Reba2.jpg” cannot be displayed, because it contains errors.

The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.

* See the remarks of Frida Saal above and of Barbara Johnson on mathematics (The Shining of May 29, cited in Readings for St. Patrick's Day).


† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005.  See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.

Thursday, March 3, 2005

Thursday March 3, 2005

Filed under: General — Tags: — m759 @ 3:26 PM
Necessity, Possibility, Symmetry

The image “http://www.log24.com/log/pix05/050303-Symmetry.gif” cannot be displayed, because it contains errors.


Matrix group actions,
March 26, 1985

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: , — m759 @ 2:20 PM

Relativity Blues

Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry.  Here is some related material.

In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).

(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)

On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:

"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."

One possible response to such statements was recently provided in some detail by a Princeton philosophy professor.  See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.

A more thoughtful response would take into account the following:

1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson

2. The physics underlying Einstein's remarks on free will, God, and dice
 
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels  Secret Passages and Broken Symmetries

4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University

5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.

From a review of Gilead by Jane Vandenburgh:  

"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"

For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries.  If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003,  and those of November 5, 2002.

From Many Dimensions (Eerdmans paperback, 1963, page 53):

"Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?"

A recent answer:

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Thursday, February 17, 2005

Thursday February 17, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

On the Lapis Philosophorum,
the Philosophers' Stone –

"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"
Many Dimensions,
by Charles Williams, 1931
(Eerdmans paperback,
April 1979, pp. 43-44)

"The lapis was thought of as a unity
and therefore often stands for
the prima materia in general."
Aion, by C. G. Jung, 1951
(Princeton paperback,
1979, p. 236)

"Its discoverer was of the opinion that
he had produced the equivalent of
the primordial protomatter
which exploded into the Universe."
The Stars My Destination,
by Alfred Bester, 1956
(Vintage hardcover,
July 1996, p. 216)

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

See also
The Diamond Archetype.

For more on modal theology, see

Kurt Gödel's Ontological Argument
and

 The Ontological Argument
 from Anselm to Gödel.

Tuesday, January 25, 2005

Tuesday January 25, 2005

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

Diamonds Are Forever

 
The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

 

Robert Stone,
A Flag for Sunrise:

" 'That old Jew gave me this here.'  Egan looked at the diamond.  'I ain't giving this to you, understand?  The old man gave it to me for my boy.  It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal.  The Boddhisattva declining nirvana out of compassion.   Contemplating the ignorance of you and me, eh?  That's a metaphor of our Buddhist friends.'

Pablo's eyes glazed over.  'Holy shit,' he said.  'Santa Maria.'  He stared at the diamond in his palm with passion.

'Hey,' he said to the priest, 'diamonds are forever!  You heard of that, right?  That means something, don't it?'

'I have heard it,' Egan said.  'Perhaps it has a religious meaning.' "
 


"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz

From

DIALECTIC AND EXISTENCE
IN KIERKEGAARD AND KANT

Nythamar Fernandes de Oliveira

Pontifical Catholic University
at Porto Alegre, Brazil

"Such is the paradoxical 'encounter' of the eternal with the temporal. Just like the Moment of the Incarnation, when the Eternal entered the temporal, Kierkegaard refers to the category of the Instant (Danish Ojeblikket, 'a glance of the eye, eyeblink,' German Augenblick) as the dialectical kernel of our existential consciousness:

If the instant is posited, so is the eternal –but also the future, which comes again like the past … The concept around which everything turns in Christianity, the concept which makes all things new, is the fullness of time, is the instant as eternity, and yet this eternity is at once the future and the past.

Although I cannot examine here the Kierkegaardian conception of time, the dialectical articulation of time and existence, as can be seen, underlies his entire philosophy of existence, just as the opposition between 'eternity' and 'temporality': the instant, as 'an atom of eternity,' serves to restructure the whole synthesis of selfhood into a spiritual one, in man’s 'ascent' toward its Other and the Unknown. In the last analysis, the Eternal transcends every synthesis between eternity and time, infinity and finiteness, preserving not only the Absolute Paradox in itself but above all the wholly otherness of God. It is only because of the Eternal, therefore, that humans can still hope to attain their ultimate vocation of becoming a Chistian. As Kierkegaard writes in Works of Love (1847),

The possibility of the good is more than possibility, for it is the eternal. This is the basis of the fact that one who hopes can never be deceived, for to hope is to expect the possibility of the good; but the possibility of the good is eternal. …But if there is less love in him, there is also less of the eternal in him; but if there is less of the eternal in him, there is also less possibility, less awareness of possibility (for possibility appears through the temporal movement of the eternal within the eternal in a human being)."

Monday, January 24, 2005

Monday January 24, 2005

Filed under: General — Tags: — m759 @ 10:23 PM

The Diamond
of Possibility

by  Keith Allen Korcz

"We symbolize logical necessity with the box (box.gif (75 bytes)) and logical possibility with the diamond (diamond.gif (82 bytes)).  I first discuss combining negations with the box and diamond, noting that logical possibility and logical necessity are inter-definable with the help of negation: box.gif (75 bytes)p = ~diamond.gif (82 bytes)~p and diamond.gif (82 bytes)p = ~box.gif (75 bytes)~p. Thus, possibility and necessity are two sides of the same coin."

And what do we           
 symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

Monday January 24, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:45 PM

Old School Tie

From a review of A Beautiful Mind:

"We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "

The image “http://www.log24.com/log/pix05/050124-Tie.gif” cannot be displayed, because it contains errors.

"Three readings of diamond and box
have been extremely influential."

Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke

"Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections….

diamonds and boxes are upper and lower adjoints of Galois connections…."

— "Modal Kleene Algebra
and Applications: A Survey"
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
 
See also
Galois Correspondence

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

and Log24.net, May 20, 2004:

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

Thursday, July 31, 2003

Thursday July 31, 2003

Filed under: General — Tags: — m759 @ 6:41 PM

Killer Radio

"See the girl with the diamond ring?
 She knows how to shake that thing."

— Jerry Lee "Killer" Lewis on
    KHYI 95.3 FM, Plano, Texas,
    at about 5:12 PM EDT 7/31/03,
    introduced by DJ Allen Peck Sr.

"And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."

— Blaise Pascal, De l'Esprit Géométrique

"When the light came she was sitting on the bed beside an open suitcase, toying with her diamond rings.  She saw the light first in the depths of the largest stone."

— Paul Preuss, Broken Symmetries,
    scene at Diamond Head, Oahu, Hawaii

Now playing (6:41 PM EDT) on Killer Radio:

"Jack of Diamonds, that's
 a hard card to find."

"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond…."

— Gerard Manley Hopkins, Society of Jesus

Perhaps Sam Phillips was twanged by a Hawaiian guitar. (See previous two entries.)

 

The Big Time

"The place outside the cosmos where I and my pals do our nursing job I simply call the Place.  A lot of my nursing consists of amusing and humanizing Soldiers fresh back from raids into time. In fact, my formal title is Entertainer…."

The Big Time,
    by Fritz Leiber

A Story That Works

  • "There is the dark, eternally silent, unknown universe;
  • there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles —
    • that minds should really touch, or
    • that the silent universe should speak, tell minds a story, or (perhaps the same thing)
    • that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
  • and lastly, there is lonely, story-telling, wonder-questing, mortal me."

    Fritz Leiber in "The Button Molder"

 

 

See also "Top Ten Most Overheard Comments by new KHYI listeners" at Miss Lana's Anything Page, entry for

Pearl Harbor Day, December 7, 2002.
 

Wednesday, July 23, 2003

Wednesday July 23, 2003

Filed under: General,Geometry — Tags: — m759 @ 4:17 PM

Being Pascal Sauvage

Pascal

"Voilà ce que je sais par une longue expérience de toutes sortes de livres et de personnes. Et sur cela je fais le même jugement de ceux qui disent que les géomètres ne leur donnent rien de nouveau par ces règles, parce qu' ils les avaient en effet, mais confondues parmi une multitude d' autres inutiles ou fausses dont ils ne pouvaient pas les discerner, que de ceux qui cherchant un diamant de grand prix

Diamant

parmi un grand nombre de faux, mais qu' ils n' en sauraient pas distinguer, se vanteraient, en les tenant tous ensemble, de posséder le véritable aussi bien que celui qui, sans s' arrêter à ce vil amas, porte la main sur la pierre choisie que l' on recherche, et pour laquelle on ne jetait pas tout le reste."

— Blaise Pascal, De l'Esprit Géométrique

La Pensée Sauvage

"….the crowning image of the kaleido­scope, lavishly analogized to the mythwork in a three-hundred-word iconic apotheosis that served to put the wraps on the sustained personification of “la pensée sauvage” in the figure of the bricoleur, in an argument developed across two chapters and some twenty pages in his [Claude Lévi-Strauss's] most famous book…."

— Robert de Marrais in
Catastrophes, Kaleidoscopes,
String Quartets:
Deploying the Glass Bead Game


Pascal Sauvage

Chiasmus

For more on pensée sauvage, see

"Claude Lévi-Strauss,

Chiasmus

and the Ethnographic Journey."

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

Wednesday, November 27, 2002

Wednesday November 27, 2002

Filed under: General,Geometry — Tags: — m759 @ 11:30 PM

Waiting for Logos

Searching for background on the phrase "logos and logic" in yesterday's "Notes toward a Supreme Fact," I found this passage:

"…a theory of psychology based on the idea of the soul as the dialectical, self-contradictory syzygy of a) soul as anima and b) soul as animus. Jungian and archetypal psychology appear to have taken heed more or less of only one half of the whole syzygy, predominantly serving an anima cut loose from her own Other, the animus as logos and logic (whose first and most extreme phenomenological image is the killer of the anima, Bluebeard). Thus psychology tends to defend the virginal innocence of the anima and her imagination…"

— Wolfgang Giegerich, "Once More the Reality/Irreality Issue: A Reply to Hillman's Reply," website 

The anima and other Jungian concepts are used to analyze Wallace Stevens in an excellent essay by Michael Bryson, "The Quest for the Fiction of an Absolute." Part of Bryson's motivation in this essay is the conflict between the trendy leftist nominalism of postmodern critics and the conservative realism of more traditional critics:

"David Jarraway, in his Stevens and the Question of Belief, writes about a Stevens figured as a proto-deconstructionist, insisting on 'Steven's insistence on dismantling the logocentric models of belief' (311) in 'An Ordinary Evening in New Haven.' In opposition to these readings comes a work like Janet McCann's Wallace Stevens Revisited: 'The Celestial Possible', in which the claim is made (speaking of the post-1940 period of Stevens' life) that 'God preoccupied him for the rest of his career.'"

Here "logocentric" is a buzz word for "Christian." Stevens, unlike the postmodernists, was not anti-Christian. He did, however, see that the old structures of belief could not be maintained indefinitely, and pondered what could be found to replace them. "Notes toward a Supreme Fiction" deals with this problem. In his essay on Stevens' "Notes," Bryson emphasizes the "negative capability" of Keats as a contemplative technique:

"The willingness to exist in a state of negative capability, to accept that sometimes what we are seeking is not that which reason can impose…."

For some related material, see Simone Weil's remarks on Electra waiting for her brother Orestes. Simone Weil's brother was one of the greatest mathematicians of the past century, André Weil.

"Electra did not seek Orestes, she waited for him…"

— Simone Weil

"…at the end, she pulls it all together brilliantly in the story of Electra and Orestes, where the importance of waiting on God rather than seeking is brought home forcefully."

— Tom Hinkle, review of Waiting for God

Compare her remarks on waiting for Orestes with the following passage from Waiting for God:

"We do not obtain the most precious gifts by going in search of them but by waiting for them. Man cannot discover them by his own powers, and if he sets out to seek for them he will find in their place counterfeits of which he will be unable to discern falsity.

The solution of a geometry problem does not in itself constitute a precious gift, but the same law applies to it because it is the image of something precious. Being a little fragment of particular truth, it is a pure image of the unique, eternal, and living Truth, the very Truth that once in a human voice declared: "I am the Truth."

Every school exercise, thought of in this way, is like a sacrament.

In every school exercise there is a special way of waiting upon truth, setting our hearts upon it, yet not allowing ourselves to go out in search of it. There is a way of giving our attention to the data of a problem in geometry without trying to find the solution…."

— Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of  God"

Weil concludes the preceding essay with the following passage:

"Academic work is one of those fields containing a pearl so precious that it is worth while to sell all of our possessions, keeping nothing for ourselves, in order to be able to acquire it."

This biblical metaphor is also echoed in the work of Pascal, who combined in one person the theological talent of Simone Weil and the mathematical talent of her brother. After discussing how proofs should be written, Pascal says

"The method of not erring is sought by all the world. The logicians profess to guide to it, the geometricians alone attain it, and apart from their science, and the imitations of it, there are no true demonstrations. The whole art is included in the simple precepts that we have given; they alone are sufficient, they alone afford proofs; all other rules are useless or injurious. This I know by long experience of all kinds of books and persons.

And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."

— Blaise Pascal, The Art of Persuasion

 

For more diamond metaphors and Jungian analysis, see

The Diamond Archetype.

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