Log24

Monday, September 1, 2014

Mathematics, Not Theology

Filed under: General — m759 @ 5:00 PM

(Continued)

“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”

— Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art
 , Cornell U. Press, 1975

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

“There is  such a thing as a three-set.”
— Saying adapted from a novel by Madeleine L’Engle

Thursday, August 3, 2017

Poetic Theology at the New York Times

Filed under: General — Tags: — m759 @ 12:19 PM

Or:  Trinity Test Site

From the New York Times Book Review  of
next Sunday, August 6, 2017 —

"In a more conventional narrative sequence,
even a sequence of poems,
this interpenetration would acquire
sequence and evolution." [Link added.]

The concept under review is that of the Holy Trinity.

See also, in this  journal, Cube Trinity.

For a simpler Trinity model, see the three-point line  

Monday, January 2, 2017

Constructivist Theology

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

This journal on December 24, 2016 (Christmas Eve)
quoted some remarks on "constructivism" in art and 
added a link to the same word as applied in mathematics:

"The word 'constructivism' also refers to
a philosophy of mathematics. See a Log24 post,
'Constructivist Witness. . . ."

From that post

From a post later the same day, Dec. 22— "The Laugh-Hospital"—

Constructivism in mathematics and the laughing academy

This  (Jan. 2, 2017) post was suggested by the reported Christmas Eve death
of a Jesuit priest, Joseph Fitzmyer.

Those entertained by the thought of constructivist laugh-hospitals may
contemplate the New Year's  Eve death of a sitcom actor who played 
a priest. See today's previous post, Sitcom Theology.

Related material — "Laugh Track" in this journal.

Sitcom Theology

Filed under: General,Geometry — Tags: — m759 @ 1:20 PM

The Hollywood Reporter

"William Christopher, best known for playing Father Mulcahy
on the hit sitcom M*A*S*H , died Saturday [Dec. 31, 2016] of
lung cancer, his agent confirmed to The Hollywood Reporter.
He was 84.

Christopher died at his home in Pasadena, with his wife by
his bedside, at 5:10 a.m. on New Year's Eve, according to a
statement from his agent."

— 5:59 PM PST 12/31/2016 by Meena Jang

Image reshown in this journal on the midnight (Eastern time)
preceding Christopher's death —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Related material —

From a Log24 search for "Deathly Hallows" —

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows symbol—
Two blocks short of  a design.

Those who prefer Latin with their theology
may search this journal for "In Nomine Patris."

Thursday, September 25, 2014

Theology and Art

Filed under: General — m759 @ 12:00 PM

Recommended reading for Josefine Lyche:

See also Ayn Sof (Jan. 7, 2011).

Wednesday, March 21, 2012

Digital Theology

Filed under: General,Geometry — Tags: , — m759 @ 7:20 AM

See also remarks on Digital Space and Digital Time in this journal.

Such remarks can, of course, easily verge on crackpot territory.

For some related  pure  mathematics, see Symmetry of Walsh Functions.

Sunday, June 2, 2019

Coordinatizing the Deathly Hallows

Filed under: General — m759 @ 10:59 PM

See as well, in this journal, Deathly HallowsRelativity Problem, and Space Cross.

A related quote "This is not mathematics; this is theology."

Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: — m759 @ 6:45 PM

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .

 .

Saturday, April 7, 2018

Dot

Filed under: General — Tags: — m759 @ 10:41 PM

The late Philip J. Davis in his 2004 essay 

"A Brief Look at Mathematics and Theology,"
Humanistic Mathematics Network Journal ,
Issue 27, Article 14. Available at:
http://scholarship.claremont.edu/hmnj/vol1/iss27/14/ 

wrote —

"In my childhood, the circle persisted as a potent magic figure
in the playtime doggerel 'Make a magic circle and sign it with a dot.'
The interested reader will find thousands of allusions to the phrase
'magic circle' on the Web."

There are fewer allusions to "magic circle" + "sign it with a dot."

One such allusion (click to enlarge) is . . .

Davis died on Pi Day .

Saturday, February 17, 2018

The Binary Revolution

Filed under: General,Geometry — Tags: , — m759 @ 5:00 PM

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • "The digital revolution based on the 2 symbols (0,1)"
  • "The algebra of George Boole"
  • "Binary codes"
  • "Dirac's spinors, with their up/down dichotomy"

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today's earlier post God's Dice and Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

Wednesday, August 9, 2017

Stability

Filed under: General — m759 @ 2:35 PM

"She wrote her doctoral thesis, which was supervised
by Friedrichs, on the stability of a spherical implosion
and was awarded her Ph.D. in 1951."

MacTutor

See also a related Google Image Search.

For images from the reported date of Morawetz's death,
see Theology for Child Buyers.

Update of 2:56 PM ET Friday, August 11, 2017 —

Legacy.com and NYU now report that Morawetz died
on Tue., Aug. 8, not, as the AMS reported, on Mon., Aug. 7.
(The AMS has now corrected its error.)

For sloppiness about mathematics that echoes this
sloppiness about dates, see a post of Tue., Aug. 8.

Thursday, August 3, 2017

Glitter at the Dark Tower

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

"The centre of transformations that
Transform for transformation's self,

In a glitter that is a life, a gold
That is a being, a will, a fate."

— Wallace Stevens, "Human Arrangement"

From "The Dark Tower," a post of July 9, 2016 —

See also a search for Glitter in this journal.

Monday, June 26, 2017

Four Dots

Analogies — "A : B  ::  C : D"  may be read  "A is to B  as  C is to D."

Gian-Carlo Rota on Heidegger…

"… The universal as  is given various names in Heidegger's writings….

The discovery of the universal as  is Heidegger's contribution to philosophy….

The universal 'as' is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger."

— "Three Senses of 'A is B' in Heideggger," Ch. 17 in Indiscrete Thoughts
 

See also Four Dots in this journal. 

Some context:  McLuhan + Analogy.

Friday, August 19, 2016

Princeton University Press in 1947

Filed under: General — Tags: , , — m759 @ 11:17 AM

From a review, in the context of Hollywood, of a Princeton
University Press book on William Blake from 1947 —

Wednesday, August 10, 2016

Analogies Test

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

Obituary for Wilford Stanton Miller, author in 1926
of the Miller Analogies Test  —  
         

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

A line for W. S. Miller, taken from "Annie Hall" —

"You know nothing of my work."

Thursday, June 2, 2016

Bullshit Studies

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

(Continued)

"The allusion to 'the most precious square of sense' shows
Shakespeare doing an almost scholastic demonstration of
the need for a ratio and interplay among the senses as
the very constitution of rationality."

— Marshall McLuhan, The Gutenberg Galaxy ,
University of Toronto Press, 1962, page 13

"What Shakespeare refers to in Lear  as the 'precious
square of sense' probably has reference to the traditional
'square of opposition' in logic and to that four-part analogy
of proportionality which is the interplay of sense and reason."     

— McLuhan, ibid. , page 241

This is of course nonsense, and, in view of McLuhan's pose
as a defender of the Catholic faith, damned  nonsense.

Epigraph by McLuhan —

"The Gutenberg Galaxy  develops a mosaic or field
approach to its problems."

I prefer a different "mosaic or field" related to the movable
blocks  of Fröbel, not the movable type  of Gutenberg.

Sunday, May 29, 2016

The Ideogram Principle …

Filed under: General,Geometry — Tags: , — m759 @ 4:23 PM

According to McLuhan

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)

A relation in four terms

A : B  ::  C : D   as   Model : Crutch  ::  Metaphor : Ornament —

See also Dueling Formulas and Symmetry.

Tuesday, July 7, 2015

Haiku for DeLillo*

Filed under: General — Tags: — m759 @ 4:23 PM

A music video that opens with remarks by Lawrence Ferlinghetti
at the Last Waltz concert (Thanksgiving Day, Nov. 25, 1976):

"Our Father, whose art's in heaven…" —

For other religious remarks from the above upload date,
Sept. 9, 2011, see Holy Field GF(3).

Click the above "ripple" image for a Grateful Dead haiku
quoted here on Sunday, July 5, 2015.

For another meditation from the second upload date above,
March 19, 2012, see some thoughts on the word "field."

IMAGE- Japanese character for 'field'

* For the title, see an excerpt from Point Omega .

Wednesday, September 17, 2014

The Horse

Filed under: General — m759 @ 10:23 AM

A New York Times  piece today on author Donald Antrim:

“The next project is a novel ‘about’ (having loosely to do with)
his father, Harry, a T. S. Eliot scholar who wrote a well-regarded
monograph on the poet.”

— John Jeremiah Sullivan

From Harry T. Antrim’s 1967 thesis on Eliot:

“That words can be made to reach across the void
left by the disappearance of God (and hence of all
Absolutes) and thereby reestablish some basis of
relation with forms existing outside the subjective
and ego-centered self has been one of the chief
concerns of the first half of the twentieth century.”

An epigraph selected by Sullivan for a 2002 Harper’s Magazine
article, “Horseman, Pass By“—

Far back, far back in our dark soul
the horse prances.

— D. H. Lawrence

A related image from pure mathematics
(a source of Absolutes unrelated to theology):

See April 9, 2004, for a post on the “Horseman” article.

Tuesday, August 12, 2014

Monkey Business

Filed under: General — m759 @ 12:00 PM

Welcome to the Ape Stuff.

Thursday, June 26, 2014

The McLuhan Dimension

Filed under: General,Geometry — Tags: , — m759 @ 2:56 AM

"History is a deep and complicated puzzle—
especially when it involves more dimensions than time."

Introduction to a novella in Analog Science Fiction

IMAGE- Marshall McLuhan

IMAGE- Annenberg Hall at Harvard

IMAGE- Search for 'quilt geometry' yields a result from Annenberg Media.

"Annenberg Hall" at Harvard was originally part of a memorial for
Civil War dead. Formerly "Alumni Hall," it was renamed in 1996.

Wednesday, May 21, 2014

Through the Vanishing Point*

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

Marshall McLuhan in "Annie Hall" —

"You know nothing of my work."

Related material — 

"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"

— Paul Simon

It was a dark and stormy night…

http://www.log24.com/log/pix11/110420-DarkAndStormy-Logicomix.jpg

— Page 180, Logicomix

A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—

IMAGE- Whitehead on Fano's construction of the 15-point projective Galois space over GF(2)

See also Absolute Ambition (Nov. 19, 2010).

* For the title, see Vanishing Point in this journal.

Friday, November 1, 2013

Cameron’s Group Theory Notes

Filed under: General,Geometry — Tags: , — m759 @ 7:00 AM

In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—

  • Definition 1.3.1, Group actions, and example on partitions of a 4-set, p. 19.
  • Exercise 1.1, The group of Fano-plane symmetries, p. 35.
  • Exercise 2.17, The group of the empty set and the 15 two-subsets of a six-set, p. 66.
  • Section 3.1.2, The holomorph of a group, p. 70.
  • Exercise 3.7, The groups A8 and AGL(4,2), p. 78.

Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.

But note also Log24 posts tagged Not Theology.

Monday, October 14, 2013

Up and Down

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

Heraclitus, Fragment 60 (Diels number):

The way up and the way down is one and the same.

ὁδὸς ἄνω κάτω μία καὶ ὡυτή

hodòs áno káto mía kaì houté

— http://www.heraclitusfragments.com/B60/index.html

IMAGE- Fetzer on ambiguity in Mann's 'Doctor Faustus'

See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Further context:  Not Theology.

Monday, October 7, 2013

Post-Production (continued)

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

This journal on Oct. 2, the date of death for
the developer of mathematical Braille —

Clicking on the image of St. Peter's Square in that post led to

Braille, as noted in last midnight's post, is based
on a six-dot cell. For some pure mathematics of
the six-dot cell, see 

Modeling the 21-point plane
with outer automorphisms of S6

Two quotations that seem relevant —

"When Death tells a story, you really have to listen"
Cover of The Book Thief

"This is not theology, this is mathematics."
Steven H. Cullinane, Sept. 22, 2013

Sunday, September 22, 2013

Incarnation, Part 2

Filed under: General,Geometry — Tags: , , — m759 @ 10:18 AM

From yesterday —

"…  a list of group theoretic invariants
and their geometric incarnation…"

David Lehavi on the Kummer 166 configuration in 2007

Related material —

IMAGE- 'This is not mathematics; this is theology.' - Paul Gordan

"The hint half guessed, the gift half understood, is Incarnation."

T. S. Eliot in Four Quartets

"This is not theology; this is mathematics."

— Steven H. Cullinane on  four quartets

To wit:


Click to enlarge.

Thursday, June 13, 2013

Gate

Filed under: General,Geometry — Tags: , — m759 @ 2:13 PM

"Eight is a Gate." — Mnemonic rhyme

Today's previous post, Window, showed a version
of the Chinese character for "field"—

This suggests a related image

The related image in turn suggests

Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.

Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.

Jan Krikke

As was the I Ching.  A related structure:

Monday, May 21, 2012

Brightness at Noon

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

Occultation according to McLuhan

Marshall McLuhan writing to Ezra Pound  on Dec. 21, 1948—

"The American mind is not even close to being amenable to the ideogram principle as yet.  The reason is simply this. America is 100% 18th Century. The 18th century had chucked out the principle of metaphor and analogy— the basic fact that as A is to B so is C to D.  AB:CD.   It can see AB relations.  But relations in four terms are still verboten.  This amounts to deep occultation of nearly all human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  Until stated and publicly recognized for what it is, poetry and the arts can’t exist in America."

For context, see Cameron McEwen, "Marshall McLuhan, John Pick, and Gerard Manley Hopkins." (Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)

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.

Tuesday, March 27, 2012

Literary Field

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

An image suggested by Google's observance today
of Mies van der Rohe's 126th birthday—

Related material:

See also yesterday's Chapter and Verse  by Stanley Fish,
and today's Arts & Letters Daily .

Tuesday, March 20, 2012

Field (continued)

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

In memory of director Ulu Grosbard (continued from yesterday)

From  http://scripturetext.com/matthew/13-44.htm —

Again the kingdom of heaven is like unto treasure hid in a field
the which when a man hath found he hideth and for joy thereof
goeth and selleth all that he hath and buyeth that field

ΚΑΤΑ ΜΑΤΘΑΙΟΝ 13:44 Greek NT: Byzantine/Majority Text (2000)
παλιν ομοια εστιν η βασιλεια των ουρανων θησαυρω κεκρυμμενω εν τω αγρω

LEXICON  


παλιν  adverb


palin  pal'-in:  (adverbially) anew, i.e. (of place) back, (of time) once more, or (conjunctionally) furthermore or on the other hand — again.


ομοια  adjective – nominative singular feminine


homoios  hom'-oy-os:  similar (in appearance or character) — like, + manner.


εστιν  verb – present indicative – third person singular 


esti  es-tee':  he (she or it) is; also (with neuter plural) they are


η  definite article – nominative singular feminine


ho  ho:  the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.


βασιλεια  noun – nominative singular feminine


basileia  bas-il-i'-ah:  royalty, i.e. (abstractly) rule, or (concretely) a realm — kingdom, + reign.


των  definite article – genitive plural masculine


ho  ho:  the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.


ουρανων  noun – genitive plural masculine


ouranos  oo-ran-os':  the sky; by extension, heaven (as the abode of God); by implication, happiness, power, eternity; specially, the Gospel (Christianity) — air, heaven(-ly), sky.


θησαυρω  noun – dative singular masculine


thesauros  thay-sow-ros':  a deposit, i.e. wealth — treasure.


κεκρυμμενω  verb – perfect passive participle – dative singular masculine 


krupto  kroop'-to:  to conceal (properly, by covering) — hide (self), keep secret, secret(-ly).


εν  preposition


en  en:  in, at, (up-)on, by, etc.


τω  definite article – dative singular masculine


ho  ho:  the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.


αγρω  noun – dative singular masculine


agros  ag-ros':  a field (as a drive for cattle); genitive case, the country; specially, a farm, i.e. hamlet — country, farm, piece of ground, land.

Monday, March 19, 2012

Field

Filed under: General,Geometry — Tags: , — m759 @ 9:08 AM

IMAGE- Two versions of 'field'

— Illustration by Neill Cameron for his father, combinatorialist Peter J. Cameron

Illustration by Nao of the Japanese (and Chinese) character for "field"—

IMAGE- Japanese character for 'field'

Related material—

Finitegeometry.org favicon from February 24, 2012

Saturday, February 18, 2012

Logo

Filed under: General,Geometry — Tags: — m759 @ 8:48 AM

Pentagram design agency on the new Windows 8 logo

"… the logo re-imagines the familiar four-color symbol
as a modern geometric shape"—

http://www.log24.com/log/pix12/120218-Windows8Logo.jpg

Sam Moreau, Principal Director of User Experience for Windows,
yesterday—

On Redesigning the Windows Logo

"To see what is in front of one's nose
needs a constant struggle."
George Orwell

That is the feeling we had when Paula Scher
(from the renowned Pentagram design agency)
showed us her sketches for the new Windows logo.

Related material:

http://www.log24.com/log/pix12/120218-SmallSpaces-256w.gif

Friday, September 16, 2011

Art, Jung, Toronto

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

http://www.log24.com/log/pix11B/110916-ArtJungToronto-NYT.jpg

Related art—

Midnight's Icons,

http://www.log24.com/log/pix11B/110915-Jung-FourDiamonds.gif

Faust in Toronto,

http://www.log24.com/log/pix11B/110916-NYT-Faust.jpg

and…

McLuhan 'tetrad' figure with four diamonds surrounding a fifth, the medium

McLuhan and the Four Diamonds

(Wikipedia figure)

"Visually, a tetrad can be depicted as
 four diamonds forming an X…."

" X never, ever, marks the spot."

Friday, September 9, 2011

Galois vs. Rubik

Filed under: General,Geometry — Tags: , , — m759 @ 2:56 PM

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from "Bleu"

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

A Beginning

Filed under: General — Tags: , — m759 @ 5:29 AM
 

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.

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: — m759 @ 6:29 PM

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

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

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

The Eightfold Cube

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

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

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

Borovik has a such a diagram—

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

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

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

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

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

Happy St. Augustine's Day.

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

Sunday, August 14, 2011

Sunday Review

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

The Sunday New York Times  today—

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

This suggests…

The Elusive Small Idea—

Part I:

McLuhan and the Seven Snow Whites

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

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

"Time for you to see the field."

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

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

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

Friday, July 29, 2011

Marshall, Meet Bagger

Filed under: General — Tags: — m759 @ 8:28 AM

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable to the ideogram principle as yet.  The reason is simply this.  America is 100% 18th Century.  The 18th century had chucked out the principle of metaphor and analogy— the basic fact that as A is to B so is C to D.  AB:CD.  It can see AB relations.  But relations in four terms are still verboten.  This amounts to deep occultation of nearly all human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  Until stated and publicly recognized for what it is, poetry and the arts can’t exist in America."

"Time for you to see the field." —Bagger Vance

The field — See June 2010.

Monday, July 11, 2011

Accentuate the Positive

Filed under: General,Geometry — Tags: — m759 @ 2:02 PM

An image that may be viewed as
a cube with a "+" on each face—

http://www.log24.com/log/pix11B/110711-EightfoldCube.gif

The eightfold cube

http://www.log24.com/log/pix11B/110711-CubeHypostases.gif

Underlying structure

For the Pope and others on St. Benedict's Day
who prefer narrative to mathematics

Friday, November 19, 2010

Absolute Ambition

Filed under: General — Tags: , — m759 @ 10:30 AM

"It's my absolute ambition that you are touched to the core of your being with the content…."

— Julie Taymor on Spider-Man: Turn Off the Dark  (Playbill video, undated)

Another ambitious comic-book promotion —

"What Logicomix  does that few works in any medium do is to make intellectual passion palpable. That is its greatest strength. And it’s here that its form becomes its substance."

— Judith Roitman, review (pdf, 3.7 MB) of Logicomix: An Epic Search for Truth , in …

http://www.log24.com/log/pix10B/101119-AMSnoticesSm.jpg

 The December 2010 AMS Notices  cover has excerpts from Logicomix.

Related material:

"In the classical grammarians’ sense of the power of form over 'content' and style over 'substance,' he originated the phrase, 'the medium is the message.'"

— Joseph P. Duggan on Marshall McLuhan at The University Bookman

See also, in this  journal, The Medium is the Message, Wechsler, and Blockheads .

Saturday, June 26, 2010

Plato’s Logos

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

“The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was ‘The Idea of the Beautiful in Plato’s Dialogues’…. My investigation traces the historical destiny of the same concept….”

* See Cassirer’s Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.

— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924

On a figure from Plato’s Meno

IMAGE- Plato's diamond and finite geometry

The above figures illustrate Husserl’s phrase  “eidetic variation”
a phrase based on Plato’s use of eidos, a word
closely related to the word “idea” in Panofsky’s title.

For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his “The Concept of Group and the Theory of Perception.”

Sketch of some further remarks—

http://www.log24.com/log/pix10A/100626-Theories.jpg

The Waterfield question in the sketch above
is from his edition of Plato’s Theaetetus
(Penguin Classics, 1987).

The “design theory” referred to in the sketch
is that of graphic  design, which includes the design
of commercial logos. The Greek  word logos
has more to do with mathematics and theology.

“If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity.”

— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato’s Theaetetus
, SUNY Press, 1990

Related material—

(Click to enlarge.)

http://www.log24.com/log/pix10A/100626-CrossOnSocratesSm.gif

Saturday, April 24, 2010

Go Ask Alice

Filed under: General — Tags: — m759 @ 1:13 PM

McLuhan in Space  by Richard Cavell—

As the word "through" in the title of Through the Vanishing Point hints… key reference points for McLuhan and Parker in writing Through the Vanishing Point  were the "Alice" books.

[The footnote symbol here is mine.]

Alice Rae, McLuhan's Unconscious, doctoral dissertation, School of History and Politics, University of Adelaide, May 2008

What McLuhan calls the "unconscious"' is more often named by him as Logos, "acoustic space" or the "media environment," and I trace the debts that these concepts owe not only to Freud and Jung, but to Aristotle, St. Thomas Aquinas, gestalt theory, art theory, Henri Bergson, Pierre Teilhard de Chardin, Wyndham Lewis, Siegfried Giedion, Harold Innis, the French symbolist poets of the late nineteenth century and the British modernists of the early twentieth.

The declaration section of the thesis is dated November 19, 2008.

Related material— Halloween 2005 and The Gospel According to Father Hardon.

A work suggested by Ander Monson's new Vanishing Point . (See April 17 and April 23, together with the April 22 picture of a non-Euclidean  point in the context of "The Seventh Symbol.")

Tuesday, March 16, 2010

Gameplayers of the Academy

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

New Game

In memory of a Jesuit who died on February 22 (see yesterday's "For the Ides of March")–

“The Game in the Ship cannot be approached as a job, a vocation, a career, or a recreation. To the contrary, it is Life and Death itself at work there. In the Inner Game, we call the Game Dhum Welur, the Mind of God."

— M. A. Foster, The Gameplayers of Zan

"… for Othello, no less than his creator Shakespeare, death without speechmaking is almost unthinkable."

"Walter Ong," by Jeet Heer (Book & Culture, July/August 2004)

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

— Jesuit quote at David Lavery's weblog today

See also this journal on February 22, the date of the Jesuit death. A post on that date mentions Ong and his teacher McLuhan, and displays a McLuhan figure related to the "joke" quote above–

McLuhan 'tetrad' figure with four diamonds surrounding a fifth, the medium

Click figure for background.

Ong discussed "agonistic" culture.
See "Sunday's Theater" and a film
based on the novel discussed there–

Menin... First line, in Greek, of the Iliad

Classics 101

IMAGE- Anthony Hopkins in 'The Human Stain'

Prof. Coleman Silk introduces
freshmen to academic values

For academic gameplayers who prefer
less emotionally challenging subjects,
there is Othello Online —

http://www.log24.com/log/pix10/100316-NewGame.jpg

"New Game. You May Pass for White to Start."

Monday, February 22, 2010

Annals of Philosophy

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

The Medium is the Message

http://www.log24.com/log/pix10/100222-McLuhan.jpg
Marshall McLuhan

From the Wikipedia article
on Marshall McLuhan–

McLuhan 'tetrad' figure with four diamonds surrounding a fifth, the medium

From yesterday

(Click images for some background.)

Ian McKellen at 'Neverwas' diamond windows

Related material:

Feast of St. Louis, 2003,

a web page on McLuhan's
student Walter J. Ong, S. J.,

and Jung and the Imago Dei

Thursday, July 16, 2009

Thursday July 16, 2009

Filed under: General — m759 @ 4:00 PM

The White Itself

David Ellerman has written that

"The notion of a concrete universal occurred in Plato's Theory of Forms [Malcolm 1991]."

A check shows that Malcolm indeed discussed this notion ("the Form as an Ideal Individual"), but not under the name "concrete universal."

See Plato on the Self-Predication of Forms, by John Malcolm, Oxford U. Press, 1991.

From the publisher's summary:

"Malcolm…. shows that the middle dialogues do indeed take Forms to be both universals and paradigms…. He shows that Plato's concern to explain how the truths of mathematics can indeed be true played an important role in his postulation of the Form as an Ideal Individual."

Ellerman also cites another discussion of Plato published by Oxford:

Kneale and Kneale on Plato's theory of forms and 'the white itself'

For a literary context, see W. K. Wimsatt, Jr., "The Structure of the Concrete Universal," Ch. 6 in Literary Theory: An Anthology, edited by Julie Rivkin and Michael Ryan, Wiley-Blackwell, 2004.

Other uses of the phrase "concrete universal"– Hegelian and/or theological– seem rather distant from the concerns of Plato and Wimsatt, and are best left to debates between Marxists and Catholics. (My own sympathies are with the Catholics.)

Two views of "the white itself" —

 "So did God cause the big bang?
 Overcome by metaphysical lassitude,
 I finally reach over to my bookshelf
 for The Devil's Bible.
 Turning to Genesis I read:
 'In the beginning
 there was nothing.
 And God said,
 'Let there be light!'
 And there was still nothing,
 but now you could see it.'"
 
 -- Jim Holt, Big-Bang Theology,
    Slate's "High Concept" department 
 
   Fiat Lux, and After

"The world was warm and white when I was born:
Beyond the windowpane the world was white,
A glaring whiteness in a leaded frame,
Yet warm as in the hearth and heart of light."

-- Delmore Schwartz

Wednesday, February 4, 2009

Wednesday February 4, 2009

Filed under: General — Tags: — m759 @ 1:23 PM
Overkill
 
In memory of
James Joyce and of
Patrick McGoohan.
who both died on
a January 13th —
Scene from 'The Seventh Seal' on McLuhan book cover
Baby Blues cartoon on global positioning systems

 

Related material:

The phrase
"Habitat Global Village"
in the previous entry.

Marshall McLuhan was
apparently the originator
of the phrase
"global village."

The phrase, coined by McLuhan,
 a Catholic, should be associated
more with Rome than
with Americus, Georgia.

"The association is the idea."
— Ian Lee, The Third Word War

Number Six meets Global Village
 

Wednesday, January 28, 2009

Wednesday January 28, 2009

Filed under: General — Tags: , — m759 @ 7:59 AM
ACTUAL BEING
continued from
October 25, 2008

John Updike at Boston Public Library, 2006, photo by Robert Spencer for The New York Times
 

"The only wealth he bestowed on his subjects lay in the richness of his descriptive language, the detailed fineness of which won him comparisons with painters like Vermeer and Andrew Wyeth."

Christopher Lehmann-Haupt in today's International Herald Tribune  

 

"These people have discovered how to turn dreams into reality. They know how to enter their dream realities. They can stay there, live there, perhaps forever."

— Alfred Bester on the inmates of Ward T in his 1953 short story, "Disappearing Act"

Related material:
"Is Nothing Sacred?"
 

 

When?

Going to dark bed there was a square round Sinbad the Sailor roc's auk's egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.

Where?

Black disc from end of Ch. 17 in Ulysses

Ulysses, conclusion of Episode 17

 

Cover of 'Through the Vanishing Point,' by Marshall McLuhan and Harley Parker

Happy Feast of
St. Thomas Aquinas.

Thursday, October 30, 2008

Thursday October 30, 2008

Filed under: General,Geometry — m759 @ 12:25 PM

Readings for
Devil's Night

Pope Benedict XVI, formerly the modern equivalent of The Grand Inquisitor

1. Today's New York Times  review
    of Peter Brook's production of
   "The Grand Inquisitor"
2. Mathematics and Theology
3. Christmas, 2005
4. Cube Space, 1984-2003

Monday, August 18, 2008

Monday August 18, 2008

Filed under: General,Geometry — m759 @ 9:00 AM
The Revelation Game
Revisited

(See also Jung’s birthday.)

Google logo, Aug. 18, 2008: Dragon playing Olympic ping pong

Lotteries on
August 17,
2008
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
No belief,
no revelation

492

Chinese
Magic
Square:

4 9 2
3 5 7
8 1 6

(See below.)

Revelation
without belief

423

4/23:

Upscale
Realism:
Triangles
in Toronto

Evening
(Belief)
Belief without
revelation

272

Rahner
on Grace

(See below.)

Belief and
revelation

406

4/06:

Ideas
and Art

No belief, no revelation:
An encounter with “492”–

“What is combinatorial mathematics? Combinatorial mathematics, also referred to as combinatorial analysis or combinatorics, is a mathematical discipline that began in ancient times. According to legend the Chinese Emperor Yu (c. 2200 B.C.) observed the magic square

4 9 2
3 5 7
8 1 6

on the shell of a divine turtle….”

— H.J. Ryser, Combinatorial Mathematics, Mathematical Association of America, Carus Mathematical Monographs 14 (1963)

Belief without revelation:
Theology and human experience,
and the experience of “272”–

From Christian Tradition Today,
by Jeffrey C. K. Goh
(Peeters Publishers, 2004), p. 438:

“Insisting that theological statements are not simply deduced from human experience, Rahner nevertheless stresses the experience of grace as the ‘real, fundamental reality of Christianity itself.’ 272

272  ‘Grace’ is a key category in Rahner’s theology.  He has expended a great deal of energy on this topic, earning himself the title, amongst others, of a ‘theologian of the graced search for meaning.’ See G. B. Kelly (ed.), Karl Rahner, in The Making of Modern Theology series (Edinburgh: T&T Clark, 1992).”

Sunday, August 3, 2008

Sunday August 3, 2008

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

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

Lucia Joyce

 

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.

 

Log 24, Sept. 3, 2003:
 
Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year's entry on this date: 

 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

 

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and


FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles
.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.
 

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.

Friday, February 1, 2008

Friday February 1, 2008

Filed under: General,Geometry — m759 @ 5:01 AM
Kindergarten Theology

On the late James Edwin Loder,
a Presbyterian minister and
a professor of Christian education
at Princeton Theological Seminary,
co-author of The Knight’s Move (1992):

“At his memorial service his daughter Tami told the story of ‘little Jimmy,’ whose kindergarten teacher recognized a special quality of mind that set him apart. ‘Every day we read a story, and after the story is over, Jimmy gets up and wants to tell us what the story means.'” — Dana R. Wright

For a related story about
knight moves and kindergarten,
see Knight Moves: The Relativity
Theory of Kindergarten Blocks
,
and Log24, Jan. 16, 17, and 18.

See also Loder’s book
(poorly written, but of some
interest in light of the above):

The Knight's Move, by Loder and Neidhardt

Opening of The Knight’s Move —

“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.

The significance of the title of this volume might stop there but for Kierkegaard’s use of the ‘knight’ image. The force of Kierkegaards’s usage might be described in relation to the chess metaphor by saying that not merely does Kierkegaard’s ‘knight of faith’ undertake a unique move within the rules of the human game, but faith transposes the whole idea of a ‘knight’s move’ into the mind of the Chess Master Himself. That is to say, chess is a game of multiple possibilities and interlocking strategies, so a chess master must combine the  continuity represented by the whole complex of the game with the unpredictable decision he must make every time it is his turn. A master chess player, then, does not merely follow the rules; in him the game becomes a construct of consciousness. The better the player the more fully the game comes into its own as a creation of human intelligence. Similarly, for Kierkegaard, the knight of faith is a unique figure in human experience. The knight shows how, by existing in faith as a creative act of Christ’s Spirit, human existence comes into its own as an expression of the mind of Christ. Thus, the ultimate form of a ‘knight’s move’ is a creative act raised to the nth power by Spiritus Creator, but it still partakes fully in the concrete pieces and patterns that comprise the nature of the human game and the game of nature.”

— James E. Loder and W. Jim Neidhardt (Helmers & Howard Publishing, 1992)

For a discussion, see Triplett’s
Thinking Critically as a Christian.”

Many would deny that such
a thing is possible; let them
read the works of T. S. Eliot.

Related material:

The Knight’s Move
discusses (badly) Hofstadter’s
“strange loop” concept; see
Not Mathematics but Theology
(Log24, July 12, 2007).

Monday, January 21, 2008

Monday January 21, 2008

Filed under: General — m759 @ 11:30 PM
Serious Numbers

"When times are mysterious
Serious numbers will always be heard."

Paul Simon

Recent events in world financial markets suggest a return to this topic, considered here on October 13, 2007.

That day's entry, on mathematics and theology, may be of use to those who are considering, as their next financial move, prayer.

Some related material:

  1. The review in the Jan. 22 New York Times of a book by mathematics vulgarizer John Allen Paulos refuting arguments for the existence of God.

  2. Arguments in a less controversial area– for and against the consistency of elementary number theory:

    FOR: Kurt Gödel, Steven H. Cullinane, and John Dawson (See Log24– Nov. 30 and Dec. 2, 2005–  and "Gödel, Inconsistency, Provability, and Truth: An Exchange of Letters" (pdf), in the American Mathematical Society Notices of April 2006.)

    AGAINST: E. B. Davies, King's College London (See above.)

  3. André Weil: "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."
     
  4. God: "605." (NY Lottery, mid-day Jan. 20, 2008) This can, of course, be interpreted as "6/05"– which is perhaps a reference to "God, the Devil, and a Bridge." Or perhaps not.

Saturday, July 14, 2007

Saturday July 14, 2007

Filed under: General — m759 @ 4:07 AM
A Note from the
Catholic University
of America


The August 2007 issue of Notices of the American Mathematical Society contains tributes to the admirable personal qualities and mathematical work of the late Harvard professor George Mackey.  For my own tributes, see Log24 on March 17, 2006April 29, 2006, and March 10, 2007.  For an entry critical of Mackey’s reductionism– a philosophical, not mathematical, error– see Log24 on May 23, 2007 (“Devil in the Details”).

Here is another attack on reductionism, from a discussion of the work of another first-rate mathematician, the late Gian-Carlo Rota of MIT:

“Another theme developed by Rota is that of ‘Fundierung.’ He shows that throughout our experience we encounter things that exist only as founded upon other things: a checkmate is founded upon moving certain pieces of chess, which in turn are founded upon certain pieces of wood or plastic. An insult is founded upon certain words being spoken, an act of generosity is founded upon something’s being handed over. In perception, for example, the evidence that occurs to us goes beyond the physical impact on our sensory organs even though it is founded upon it; what we see is far more than meets the eye. Rota gives striking examples to bring out this relationship of founding, which he takes as a logical relationship, containing all the force of logical necessity. His point is strongly antireductionist. Reductionism is the inclination to see as ‘real’ only the foundation, the substrate of things (the piece of wood in chess, the physical exchange in a social phenomenon, and especially the brain as founding the mind) and to deny the true existence of that which is founded. Rota’s arguments against reductionism, along with his colorful examples, are a marvelous philosophical therapy for the debilitating illness of reductionism that so pervades our culture and our educational systems, leading us to deny things we all know to be true, such as the reality of choice, of intelligence, of emotive insight, and spiritual understanding. He shows that ontological reductionism and the prejudice for axiomatic systems are both escapes from reality, attempts to substitute something automatic, manageable, and packaged, something coercive, in place of the human situation, which we all acknowledge by the way we live, even as we deny it in our theories.”

Robert Sokolowski, foreword to Rota’s Indiscrete Thoughts

Father Robert Sokolowski

Father Robert Sokolowski

Fr. Robert Sokolowski, Ph.D., is Professor of Philosophy at The Catholic University of America in Washington, D.C. Ordained a Roman Catholic priest in 1962, he is internationally recognized and honored for his work in philosophy, particularly phenomenology. In 1994, Catholic University sponsored a conference on his work and published several papers and other essays under the title, The Truthful and the Good, Essays In Honor of Robert Sokolowski.

Thomas Aquinas College newsletter

The tributes to Mackey are contained in the first of two feature articles in the August 2007 AMS Notices.  The second feature article is a review of a new book by Douglas Hofstadter.  For some remarks related to that article, see Thursday’s Log24 entry “Not Mathematics but Theology.”

Friday, July 13, 2007

Friday July 13, 2007

Filed under: General,Geometry — m759 @ 7:00 AM
Today’s birthday:
Harrison Ford is 65.

The image “http://www.log24.com/log/pix07/070713-Ford2.jpg” cannot be displayed, because it contains errors.

“Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found.”

— “Credences of Summer,” VII,
    by Wallace Stevens, from
    Transport to Summer (1947)

“It was Plato who best expressed– who veritably embodied– the tension between the narrative arts and mathematics….


Plato clearly loved them both, both mathematics and poetry.  But he approved of mathematics, and heartily, if conflictedly, disapproved of poetry.  Engraved above the entrance to his Academy, the first European university, was the admonition: Oudeis ageometretos eiseto.  Let none ignorant of geometry enter.  This is an expression of high approval indeed, and the symbolism could not have been more perfect, since mathematics was, for Plato, the very gateway for all future knowledge.  Mathematics ushers one into the realm of abstraction and universality, grasped only through pure reason.  Mathematics is the threshold we cross to pass into the ideal, the truly real.”

   — Rebecca Goldstein, Mathematics and the Character of Tragedy

Related material:

Previous entry,
entries of July 1, 2007,
and A Little Story
(9/30/06)

Thursday, July 12, 2007

Thursday July 12, 2007

Filed under: General — Tags: — m759 @ 7:00 PM
On Interpenetration,
or Coinherence, of Souls

The August 2007 issue of Notices of the American Mathematical Society contains a review of a new book by Douglas Hofstadter, I Am a Strange Loop. (2007, Basic Books, New York. $26.95, 412 pages.)

A better review, in the Los Angeles Times of March 18, 2007, notes an important phrase in the book, "interpenetration of souls," that the AMS Notices review ignores.

Here is an Amazon.com search on "interpenetration" in the Hofstadter book:

1. on Page 217:
"… described does not create a profound blurring of two people's identities. Tennis and driving do not give rise to deep interpenetrations of souls. …"
2. on Page 237:
"… What seems crucial here is the depth of interpenetration of souls the sense of shared goals, which leads to shared identity. Thus, for instance, Carol always had a deep, …"
3. on Page 270:
"… including the most private feelings and the most confidential confessions, then the interpenetration of our worlds becomes so great that our worldviews start to fuse. Just as I could jump to California when …"
4. on Page 274:
"… we choose to downplay or totally ignore the implications of the everyday manifestations of the interpenetration of souls. Consider how profoundly wrapped up you can become in a close friend's successes and failures, in their very …"
5. on Page 276:
"… Interpenetration of National Souls Earlier in this chapter, I briefly offered the image of a self as analogous to a country …"
6. from Index:
"… birthday party for, 350 "bachelor", elusiveness of concept, 178 bad-breath analogy, 150 bandwidth of communication as determinant of degree of interpenetration, 212 213, 220, …"
7. from Index:
"… phrases denying interpenetration of souls, 270 271; physical phenomena that lack consciousness, 281 282; physical structures lacking hereness, 283; potential personal attributes, 183; …"

The American Mathematical Society editors and reviewer seem to share Hofstadter's ignorance of Christian doctrine; they might otherwise have remembered a rather famous remark: "This is not mathematics, it is theology."
 
For more on the theology of interpenetration, see Log24 on "Perichoresis, or Coinherence" (Jan. 22, 2004).

For a more mathematical approach to this topic, see Spirituality Today, Spring 1991:

"… the most helpful image is perhaps the ellipse often used to surround divine figures in ancient art, a geometrical figure resulting from the overlapping, greater or lesser, of two independent circles, an interpenetration or coinherence which will, in some sense, reunify divided humanity, thus restoring to some imperfect degree the original image of God."

See also the trinitarian doctrine implicit in related Log24 entries of July 1, 2007, which include the following illustration of the geometrical figure described, in a somewhat confused manner, above:

The image “http://www.log24.com/log/pix07/070701-Ratio.jpg” cannot be displayed, because it contains errors.

"Values are rooted
in narrative."

Harvey Cox,    
Hollis Professor
of Divinity
at Harvard,
Atlantic Monthly,
  November 1995  

Related material:

Steps Toward Salvation:
An Examination of
Co-Inherence and
Substitution in
the Seven Novels
of Charles Williams
,
by Dennis L. Weeks

Tuesday, April 3, 2007

Tuesday April 3, 2007

Filed under: General — Tags: , — m759 @ 1:00 AM

Mathematics Awareness Month

 
Related material:

"But what is it?"
Calvin demanded.
"We know that it's evil,
but what is it?"

"Yyouu hhave ssaidd itt!"
Mrs. Which's voice rang out.
"Itt iss Eevill. Itt iss thee
Ppowers of Ddarrkknesss!"

A Wrinkle in Time

AMS Notices cover, April 2007

"After A Wrinkle in Time was finally published, it was pointed out to me that the villain, a naked disembodied brain, was called 'It' because It stands for Intellectual truth as opposed to a truth which involves the whole of us, heart as well as mind.  That acronym had never occurred to me.  I chose the name It intuitively, because an IT does not have a heart or soul.  And I did not understand consciously at the time of writing that the intellect, when it is not informed by the heart, is evil."

See also
"Darkness Visible"
in ART WARS.
 
"When all is said and done,
science is about things and
theology is about words."
— Freeman Dyson,
New York Review of Books,
issue dated May 28, 1998

"Does the word 'tesseract'
mean anything to you?"
 

Friday, March 16, 2007

Friday March 16, 2007

Filed under: General,Geometry — Tags: — m759 @ 10:48 AM
"Geometry,
 Theology,
 and Politics:

 
Context and Consequences of 

the Hobbes-Wallis Dispute"
(pdf)

 

by Douglas M. Jesseph
Dept. of Philosophy and Religion
North Carolina State University

Excerpt:

"We are left to conclude that there was something significant in Hobbes's philosophy that motivated Wallis to engage in the lengthy and vitriolic denunciation of all things Hobbesian.

In point of fact, Wallis made no great secret of his motivations for attacking Hobbes's geometry, and the presence of theological and political motives is well attested in a 1659 letter to Huygens. He wrote:

But regarding the very harsh diatribe against Hobbes, the necessity of the case, and not my manners, led to it. For you see, as I believe, from other of my writings how peacefully I can differ with others and bear those with whom I differ. But this was provoked by our Leviathan (as can be easily gathered fro his other writings, principally those in English), when he attacks with all his might and destroys our universities (and not only ours, but all, both old and new), and especially the clergy and all institutions and all religion. As if the Christian world knew nothing sound or nothing that was not ridiculous in philosophy or religion; and as if it has not understood religion because it does not understand philosophy, nor philosophy because it does not understand mathematics. And so it seemed necessary that now some mathematician, proceeding in the opposite direction, should show how little he understand this mathematics (from which he takes his courage). Nor should we be deterred from this by his arrogance, which we know will vomit poison and filth against us. (Wallis to Huygens, 11 January, 1659; Huygens 1888-1950,* 2: 296-7)

The threats that Hobbes supposedly posed to the universities, the clergy, and all religion are a consequence of his political and theological doctrines. Hobbes's political theory requires that the power of the civil sovereign be absolute and undivided. As a consequence, such institutions as universities and the clergy must submit to the dictates of the sovereign in all matters. This extends, ironically enough, to geometry, since Hobbes notoriously claimed that the sovereign could ban the teaching of the subject and order 'the burning of all books of Geometry' if he should judge geometric principles 'a thing contrary to [his] right of dominion, or to the interest of men that have dominion' (Leviathan (1651) 1.11, 50; English Works** 3: 91). In the area of church government, Hobbes's doctrines are a decisive rejection of the claims of Presbyterianism, which holds that questions of theological doctrine is [sic] to be decided by the elders of the church– the presbytery– without reference to the claims of the sovereign. As a Presbyterian minister, a doctor of divinity, and professor of geometry at Oxford, Wallis found abundant reason to reject this political theory."

* Huygens, Christiaan. 1888-1950. Les oeuvres complètes de Chrisiaan Huygens. Ed. La Société Hollandaise des Sciences. 22 vols. The Hague: Martinus Nijhoff.

** Hobbes, Thomas. [1839-45] 1966. The English Works of Thomas Hobbes of Malmesbury, now First Collected and Edited by Sir William Molesworth. Edited by William Molesworth. 11 vols. Reprint. Aalen, Germany: Scientia Verlag.

 

Related material:

"But what is it?"
Calvin demanded.
"We know that it's evil,
but what is it?"

"Yyouu hhave ssaidd itt!"
Mrs. Which's voice rang out.
"Itt iss Eevill. Itt iss thee
Ppowers of Ddarrkknesss!"

A Wrinkle in Time

The image “http://www.log24.com/log/pix07/070316-AMScover.jpg” cannot be displayed, because it contains errors.

"After A Wrinkle in Time was finally published, it was pointed out to me that the villain, a naked disembodied brain, was called 'It' because It stands for Intellectual truth as opposed to a truth which involves the whole of us, heart as well as mind.  That acronym had never occurred to me.  I chose the name It intuitively, because an IT does not have a heart or soul.  And I did not understand consciously at the time of writing that the intellect, when it is not informed by the heart, is evil."

 

See also
"Darkness Visible"
in ART WARS.
 

Wednesday, January 31, 2007

Wednesday January 31, 2007

Filed under: General — m759 @ 3:09 PM
Ontotheology

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

“It may be that universal history is the
history of the different intonations
given a handful of metaphors.”
— Jorge Luis Borges (1951),
“The Fearful Sphere of Pascal,”
in Labyrinths, New Directions, 1962

“Before introducing algebraic semiotics and structural blending, it is good to be clear about their philosophical orientation. The reason for taking special care with this is that, in Western culture, mathematical formalisms are often given a status beyond what they deserve. For example, Euclid wrote, ‘The laws of nature are but the mathematical thoughts of God.'”

— Joseph A. Goguen, “Ontology, Society, and Ontotheology” (pdf)

Goguen does not give a source for this alleged “thoughts of God” statement.

A Web search for the source leads only to A Mathematical Journey, by Stanley Gudder, who apparently also attributes the saying to Euclid.

Neither Goguen nor Gudder seems to have had any interest in the accuracy of the Euclid attribution.

Talk of “nature” and “God” seems unlikely from Euclid, a pre-Christian Greek whose pure mathematics has (as G. H. Hardy might be happy to point out) little to do with either.

Loose talk about God’s thoughts has also been attributed to Kepler and Einstein… and we all know about Stephen Hawking.

Gudder may have been misquoting some other author’s blather about Kepler.  Another possible source of the “thoughts of God” phrase is Hans Christian Oersted. The following is from Oersted’s The Soul in Nature

“Sophia. Nothing of importance; though indeed I had one question on my lips when the conversion took the last turn. When you alluded to the idea, that the Reason manifested in Nature is infallible, while ours is fallible, should you not rather have said, that our Reason accords with that of Nature, as that in the voice of Nature with ours?

Alfred. Each of these interpretations may be justified by the idea to which it applies, whether we start from ourselves or external nature. There are yet other ways of expressing it; for instance, the laws of Nature are the thoughts of  Nature.

Sophia. Then these thoughts of Nature are also thoughts of God.

Alfred. Undoubtedly so, but however valuable the expression may be, I would rather that we should not make use of it till we are convinced that our investigation leads to a view of Nature, which is also the contemplation of God. We shall then feel justified by a different and more perfect knowledge to call the thoughts of Nature those of God; I therefore beg you will not proceed to [sic] fast.”

Oersted also allegedly said that “The Universe is a manifestation of an Infinite Reason and the laws of Nature are the thoughts of God.” This remark was found (via Google book search) in an obscure journal that does not give a precise source for the words it attributes to Oersted.

The image “http://www.log24.com/log/pix07/070131-OerstedGudder.jpg” cannot be displayed, because it contains errors.

Sunday, December 31, 2006

Sunday December 31, 2006

Filed under: General — Tags: — m759 @ 8:00 PM
Tools of
Christ Church
(Continued from
St. Thomas Becket’s day)

The author of the thesis
“Conversations with the Dead”

described in this morning’s entry,

Aesthetics of Evil
vs. Christ Church
,

is Darren Joseph Danylyshen.

 
This may be the same
Darren Danylyshen who has
taught at St. Stephen’s SS
(a Catholic secondary school
in Bowmanville, Ontario).
 
Following a link in the
section of that school’s site
beneath the title
“St. Stephen’s Goes Hollywood,”
we find the following:
 
The image “http://www.log24.com/log/pix06B/061231-McLuhan.jpg” cannot be displayed, because it contains errors.
 
This ties in rather neatly with the
“Tools of Christ Church” entry
for last Friday–
St. Thomas Becket’s day–
and with the fact that
today would be the feast day
of Marshall McLuhan,
if McLuhan were a saint.
(McLuhan, a Catholic, died on
Dec. 31, 1980.)
 
Related material:
 
The Communion of Saints as
the Association of Ideas
 

Sunday, November 19, 2006

Sunday November 19, 2006

Filed under: General — m759 @ 2:02 PM

Signature

From AP’s “Today in History,” Nov. 19, 2006:

Today’s birthdays: … Actress-director Jodie Foster is 44….

Thought for Today: “My theology, briefly, is that the universe was dictated but not signed.” –[Attributed to] Christopher Morley, American author and journalist (1890-1957).

A different story: Carl Sagan, Contact, Chapter 24– “The Artist’s Signature.”

Yet another story:  The Pennsylvania lottery yesterday, November 18, 2006– mid-day 914, evening 945. For interpretations, see 9/14 (Feast of the Triumph of the Cross) and also the following “signature” (i.e., “denominator”):

The image “http://www.log24.com/log/pix06B/061119-Zeta6.jpg” cannot be displayed, because it contains errors.

Number theorists may prefer to
think of 945 as the smallest
odd abundant number
(Al-Baghdadi, ca. 980-1037).

Neither of these occurrences
 of 945 in mathematics seems
 particularly divine; perhaps there
are some other properties of
 this number that make it more
credible as a divine signature–
other, that is, than its occurrence
in a lottery just in time for
Jodie Foster’s birthday.

Friday, March 10, 2006

Friday March 10, 2006

Filed under: General — m759 @ 7:59 PM

Women’s History Month continues…

Raiders of the Lost

Stone

The image “http://www.log24.com/log/pix06/060310-Stone.jpg” cannot be displayed, because it contains errors.

In honor of the upcoming program
on Women and Mathematics
at the Institute for Advanced Study
and of Sharon Stone’s 2005 lecture
at Harvard’s Memorial Church,

here are links to reviews of
two Sharon Stone classics:


“King Solomon’s Mines” (1985),
said to be inspired by the
1981 box-office success
of
“Raiders of the Lost Ark,” and

“Diabolique” (1996), starring
Stone as
a teacher of mathematics
at St. Anselm’s School for Boys.

For related material on St. Anselm
and mathematics at Princeton, see
Modal Theology and the
April 2006 AMS Notices
on Kurt Gödel.

See also yesterday’s entry
and
Log24, Jan. 1-15, 2006.

Today’s birthdays:
Sharon Stone and
Gregory La Cava.

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)

Thursday, June 9, 2005

Thursday June 9, 2005

Filed under: General,Geometry — Tags: — m759 @ 7:45 PM
Kernel of Eternity

continued

"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

From "The Relations between
Poetry and Painting," by Wallace Stevens:

"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…. It was from the point of view of… [such a] 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."

As yesterday's entry "Kernel of Eternity" indicated, the word "kernel" has a definite meaning in mathematics.  The Klein four-group, beloved of structural anthropologists and art theorists, is a particularly apt example of a kernel. (See PlanetMath for details.)

Diagrams of this group may have influenced Giovanni Sambin, professor of mathematical logic at the University of Padua; the following impressive-looking diagram is from Sambin's

The image “http://www.log24.com/theory/images/SambinBP1Pic2A.jpg” cannot be displayed, because it contains errors.

Sambin argues that this diagram reflects some of the basic structures of thought itself… making it perhaps one way to describe what  Klee called the "mind or heart of creation." 

But this verges on what Stevens called the sacerdotal.  It seems that a simple picture of the "kernel of eternity" as the four-group, a picture without reference to logic or philosophy, and without distracting letters and labels, is required.  The following is my attempt to supply such a picture:

Klein four-group

This is a picture of the four-group
as a permutation group on four points.
Pairs of colored arrows indicate the three
transformations other than the identity,
which may be regarded either as
invisible or as rendered by
the four black points themselves.

Update of 7:45 PM Thursday:

Review of the above (see comments)
by a typical Xanga reader:

"Ur a FUCKIN' LOSER!!!!!  LMFAO!!!!"

For more merriment, see
The Optical Unconscious
and
The Painted Word.

A recent Xangan movie review:

"Annakin's an idiot, but he's not an idiot because that's the way the character works, he's an idiot because George Lucas was too lazy to make him anything else. He has to descend to the Daaaahk Side, but the dark side never really seems all that dark. He kills children, but offscreen. We never get to see the transformation. One minute he cares about the republic, the next he's killing his friends, and then for some reason he's duelling with Obi Wan on a lava flow. Who cares? Not me….

So a big ol' fuck you to George Lucas. Fuck you, George!"

Both Xangans seem to be fluent in what Tom Wolfe has called the "fuck patois."

A related suggestion from Google:

Give Dad a photo gift

These remarks from Xangans and Google
 suggest the following photo gift,
based on a 2003 journal entry:

The image “http://www.log24.com/log/pix05A/050609-Fahne.jpg” cannot be displayed, because it contains errors.

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.

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)

Saturday, July 26, 2003

Saturday July 26, 2003

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

The Transcendent
Signified

“God is both the transcendent signifier
and transcendent signified.”

— Caryn Broitman,
Deconstruction and the Bible

“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”

— Henry P. Mills,
The Significance of Language,
Footnote 2

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

Simon Blackburn, Think,
Oxford University Press, 1999, page 268

The question of universals is still being debated in Paris.  See my July 25 entry,

A Logocentric Meditation.

That entry discusses an essay on
mathematics and postmodern thought
by Michael Harris,
professor of mathematics
at l’Université Paris 7 – Denis Diderot.

A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions.  Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective.  In other words, pi can be identified algebraically with X, the variable par excellence.”

Harris illustrates this with
an X in a rectangle:

For the complete passage, click here.

If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey.  On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”

Kubrick’s
monolith

Harris’s
slab

For a more serious discussion of deconstruction and Christian theology, see

Walker Percy’s Semiotic.

Friday, July 25, 2003

Friday July 25, 2003

Filed under: General — m759 @ 5:24 PM

For Jung’s 7/26 Birthday:
A Logocentric Meditation

Leftist academics are trying to pull a fast one again.  An essay in the most prominent American mathematical publication tries to disguise a leftist attack on Christian theology as harmless philosophical woolgathering.

In a review of Vladimir Tasic’s Mathematics and the Roots of Postmodern Thought, the reviewer, Michael Harris, is being less than candid when he discusses Derrida’s use of “logocentrism”:

“Derrida uses the term ‘logocentrism’… as ‘the metaphysics of phonetic writing’….”

Notices of the American Mathematical Society, August 2003, page 792

We find a rather different version of logocentrism in Tasic’s own Sept. 24, 2001, lecture “Poststructuralism and Deconstruction: A Mathematical History,” which is “an abridged version of some arguments” in Tasic’s book on mathematics and postmodernism:

“Derrida apparently also employs certain ideas of formalist mathematics in his critique of idealist metaphysics: for example, he is on record saying that ‘the effective progress of mathematical notation goes along with the deconstruction of metaphysics.’

Derrida’s position is rather subtle. I think it can be interpreted as a valiant sublation of two completely opposed schools in mathematical philosophy. For this reason it is not possible to reduce it to a readily available philosophy of mathematics. One could perhaps say that Derrida continues and critically reworks Heidegger’s attempt to ‘deconstruct’ traditional metaphysics, and that his method is more ‘mathematical’ than Heidegger’s because he has at his disposal the entire pseudo-mathematical tradition of structuralist thought. He has himself implied in an interview given to Julia Kristeva that mathematics could be used to challenge ‘logocentric theology,’ and hence it does not seem unreasonable to try looking for the mathematical roots of his philosophy.”

The unsuspecting reader would not know from Harris’s review that Derrida’s main concern is not mathematics, but theology.  His ‘deconstruction of metaphysics’ is actually an attack on Christian theology.

From “Derrida and Deconstruction,” by David Arneson, a University of Manitoba professor and writer on literary theory:

Logocentrism: ‘In the beginning was the word.’ Logocentrism is the belief that knowledge is rooted in a primeval language (now lost) given by God to humans. God (or some other transcendental signifier: the Idea, the Great Spirit, the Self, etc.) acts a foundation for all our thought, language and action. He is the truth whose manifestation is the world.”

Some further background, putting my July 23 entry on Lévi-Strauss and structuralism in the proper context:

Part I.  The Roots of Structuralism

“Literary science had to have a firm theoretical basis…”

Part II.  Structuralism/Poststructuralism

“Most [structuralists] insist, as Levi-Strauss does, that structures are universal, therefore timeless.”

Part III.  Structuralism and
             Jung’s Archetypes

Jung’s “theories, like those of Cassirer and Lévi-Strauss, command for myth a central cultural position, unassailable by reductive intellectual methods or procedures.”

And so we are back to logocentrism, with the Logos — God in the form of story, myth, or archetype — in the “central cultural position.”

What does all this have to do with mathematics?  See

Plato’s Diamond,

Rosalind Krauss on Art –

“the Klein group (much beloved of Structuralists)”

Another Michael Harris Essay, Note 47 –

“From Krauss’s article I learned that the Klein group is also called the Piaget group.”

and Jung on Quaternity:
      Beyond the Fringe –

“…there is no denying the fact that [analytical] psychology, like an illegitimate child of the spirit, leads an esoteric, special existence beyond the fringe of what is generally acknowledged to be the academic world.”

What attitude should mathematicians have towards all this? 

Towards postmodern French
  atheist literary/art theorists –

Mathematicians should adopt the attitude toward “the demimonde of chic academic theorizing” expressed in Roger Kimball’s essay, Feeling Sorry for Rosalind Krauss.

Towards logocentric German
  Christian literary/art theorists –

Mathematicians should, of course, adopt a posture of humble respect, tugging their forelocks and admitting their ignorance of Christian theology.  They should then, if sincere in their desire to honestly learn something about logocentric philosophy, begin by consulting the website

The Quest for the Fiction of an Absolute.

For a better known, if similarly disrespected, “illegitimate child of the spirit,” see my July 22 entry.

Thursday, June 26, 2003

Thursday June 26, 2003

Filed under: General — Tags: , — m759 @ 5:00 AM

ART WARS:
Art at the Vanishing Point

From the web page Art Wars:

"For more on the 'vanishing point,'
or 'point at infinity,' see
Midsummer Eve's Dream."

On Midsummer Eve, June 23, 2003, minimalist artist Fred Sandback killed himself.

Sandback is discussed in The Dia Generation, an April 6, 2003, New York Times Magazine article that is itself discussed at the Art Wars page.

Sandback, who majored in philosophy at Yale, once said that

"Fact and illusion are equivalents."

Two other references that may be relevant:

The Medium is
the Rear View Mirror
,

which deals with McLuhan's book Through the Vanishing Point, and a work I cited on Midsummer Eve  …

Chapter 5 of Through the Looking Glass:

" 'What is it you want to buy?' the Sheep said at last, looking up for a moment from her knitting.

'I don't quite know yet,' Alice said very gently.  'I should like to look all round me first, if I might.'

'You may look in front of you, and on both sides, if you like,' said the Sheep; 'but you ca'n't look all round you — unless you've got eyes at the back of your head.'

But these, as it happened, Alice had not got: so she contented herself with turning round, looking at the shelves as she came to them.

The shop seemed to be full of all manner of curious things — but the oddest part of it all was that, whenever she looked hard at any shelf, to make out exactly what it had on it, that particular shelf was always quite, empty, though the others round it were crowded as full as they could hold.

'Things flow about so here!' she said at last in a plaintive tone…."

 "When Alice went
     through the vanishing point
"
 

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