Log24

Friday, August 20, 2021

Diamond Brackets*

Filed under: General — Tags: , , — m759 @ 9:57 am

For more on the phrase "diamond brackets," see the post
Artistic Style of July 24, 2018.

This was the dies natalis  (in the Catholic sense) of philosophy
professor Garth L. Kemerling.

From Kemerling's internet "Philosophy Pages" —

"First, it must be possible in principle to arrange and organize
the chaos of our many individual sensory images by tracing
the connections that hold among them. This Kant called
the synthetic unity of the sensory manifold.

Second, it must be possible in principle for a single subject
to perform this organization by discovering the connections
among perceived images. This is satisfied by what Kant called
the transcendental unity of apperception."

Related Log24 phrases —

"Intake Manifold" and "Bulk Apperception."

* See also Bracketing (phenomenology) in Wikipedia.

Saturday, March 7, 2015

Film and Phenomenology

Filed under: General,Geometry — m759 @ 1:18 pm

Continued from All Hallows' Eve, 2014.

Last year's Halloween post displayed the
Dürer print Knight, Death, and the Devil 
(illustrated below on the cover of the book
Film and Phenomenology  by Allan Casebier).

Cover illustration: Durer's 'Knight, Death, and the Devil'

Cover illustration: Knight, Death, and the Devil
by Albrecht Dürer

Some mathematics related to a different Dürer print —

Saturday, December 8, 2012

Defining the Contest…

Filed under: General,Geometry — Tags: , , , , , — m759 @ 5:48 am

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

"Happy Birthday, Noam Chomsky." by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

See Meno Diamond in this journal. For instance, from 
the Feast of Saint Nicholas (Dec. 6th) this year—

The Meno Embedding

Plato's Diamond embedded in The Matrix

For related truths about geometry, see the diamond theorem.

For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).

See esp. the Sept. 11 post,  on a Royal Society paper from July 2012
claiming that

"With the results presented here, we have taken the first steps
in decoding the uniquely human  fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "

The sorts of patterns discussed in the 2012 paper —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

Filed under: General — Tags: , , — m759 @ 12:34 pm
 

Cullinane Diamond Theorem Research Report

by https://you.com/?chatMode=research on March 3, 2024

Overview of the Cullinane Diamond Theorem

The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG).

The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry.

Symmetry and Group Theory

The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).

This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns.

The Diamond Theorem in Various Dimensions

The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes.

The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves.

Algebraic Implications

The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings.

This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4).

Philosophical and Cultural Context

Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau.

The theorem is seen as a correct statement of the relation between geometry and the physical universe.

Clarification on Cullinan Diamonds

It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels.

The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history.

In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry.

Instagram ad for You.com AI in research mode

"Show me ALL  your sources, babe."

— Line adapted from Leonardo DiCaprio

Monday, February 19, 2024

Theology for Sophists

Filed under: General — m759 @ 10:15 pm

"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig

"It’s all in Plato, all in Plato;
bless me, what do  they
teach them at these schools?”

— C. S. Lewis in
The Narnia Chronicles

Compare and Contrast — Plato's Diamond.

Plato's diamond in Jowett's version of the Meno dialogue

Sunday, August 6, 2023

Annals of Iconography: The Dark Center

Filed under: General — Tags: — m759 @ 11:46 am

At the National Comedy Center

“There are dark comedies. There are screwball comedies.
But there aren’t many dark screwball comedies.
And if Nora Ephron’s Lucky Numbers  is any indication,
there’s a good reason for that.”

— Todd Anthony, South Florida Sun-Sentinel 

Monday, September 14, 2020

Socrates in the Marketplace

Filed under: General — Tags: — m759 @ 7:39 am

Plato's diamond in Jowett's version of the Meno dialogue

Diamond Matrix slide template at presentationgo.com

“The 2×2 matrix is commonly used in business strategy
as a representational tool to show conflicting concepts and
for decision making. This four-quadrant matrix diagram
is perfect to be used for business or marketing matrices
like BCG, SWOT, Ansoff, risk assessment…

Additionally, it will also be suitable to illustrate 4 ideas or
concepts.” [Link on “illustrate” added.]

See also a Log24 search for “Resplendent.”

Sunday, August 2, 2020

The Sword and the Stone

Filed under: General — Tags: , , — m759 @ 12:42 pm

A post of May 26, 2005, displays, if not the sword,
a place  for it —

Drama of the Diagonal

"The beautiful in mathematics resides in contradiction.
Incommensurability, logoi alogoi, was the first splendor
in mathematics." — Simone Weil, Oeuvres Choisies,
éd. Quarto
, Gallimard, 1999, p. 100

Logos Alogos  by S. H. Cullinane

"To a mathematician, mathematical entities have their own existence,
they habitate spaces created by their intention.  They do things,
things happen to them, they relate to one another.  We can imagine
on their behalf all sorts of stories, providing they don't contradict
what we know of them.  The drama of the diagonal, of the square…"

— Dennis Guedj, abstract of "The Drama of Mathematics," a talk
to be given this July at the Mykonos conference on mathematics and
narrative. For the drama of the diagonal of the square, see

Wednesday, April 15, 2020

Oslo Prophet (after Varignon)

Filed under: General — Tags: , — m759 @ 12:06 pm

See also Invariance, a Log24 post from yesterday morning —

Note the resemblance to Plato’s Diamond.

Tuesday, April 14, 2020

Invariance

Filed under: General — Tags: , — m759 @ 9:00 am

Note the resemblance to Plato’s Diamond.

Click the Pritchard passage above for an interactive version.

Sunday, October 28, 2018

Commonwealth Tales, or “Lost in Physics”

Filed under: General — m759 @ 11:00 pm

From Ulysses , by James Joyce —

John Eglinton, frowning, said, waxing wroth:

—Upon my word it makes my blood boil to hear anyone compare Aristotle with Plato.

—Which of the two, Stephen asked, would have banished me from his commonwealth?

Compare and contrast:

Plato's diamond in Jowett's version of the Meno dialogue

Fans of Plato might enjoy tales of Narnia, but fans of
James Joyce and Edgar Allan Poe might prefer
a tale by Michael Chabon from April 2001 about a
"doleful little corner of western Pennsylvania."

Saturday, September 15, 2018

Eidetic Reduction in Geometry

Filed under: G-Notes,General,Geometry — Tags: , , , , , — m759 @ 1:23 am
 

"Husserl is not the greatest philosopher of all times.
He is the greatest philosopher since Leibniz."

Kurt Gödel as quoted by Gian-Carlo Rota

Some results from a Google search —

Eidetic reduction | philosophy | Britannica.com

Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart …

Phenomenology Online » Eidetic Reduction

The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research …

Eidetic reduction – New World Encyclopedia

Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.

Terminology: Eidos

For example —

The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between the subsquares or subcubes.

See the diamond theorem and the eightfold cube.

* Cf. posts tagged Interality and Interstice.

Monday, May 14, 2018

Logos at Harvard

Filed under: General,Geometry — Tags: , , — m759 @ 3:01 pm

In 2013, Harvard University Press changed its logo to an abstract "H."

Harvard University Press Logo, Before and After

Both logos now accompany a Harvard video first published in 2012,
"The World of Mathematical Reality." 

In the video, author Paul Lockhart discusses Varignon's theorem
without naming Varignon (1654-1722) . . .

Paul Lockhart on geometry

A related view of "mathematical reality" —

Note the resemblance to Plato's Diamond.

Saturday, April 14, 2018

Immanentizing the Transcendence

Filed under: General — Tags: , — m759 @ 10:15 am

The title refers to the previous two posts.

Related literature —

Plato's Ghost: The Modernist Transformation of Mathematics
(Princeton University Press, 2008)  and . . .

Plato's diamond-in-a-matrix:

Plato's diamond in Jowett's version of the Meno dialogue

Friday, April 6, 2018

A Service

Filed under: General — m759 @ 11:36 am

From a Boston Globe obituary for Andrew Lewis, an Oscar-nominated
screenwriter who reportedly died at 92 on Feb. 28, 2018 —

"A service has been held for Mr. Lewis . . . ."

—  Bryan Marquard, Globe staff, April 5, 2018

From this  journal on the reported date of his death —

The Globe reports that Lewis's father was Clarence Irving Lewis,
a professor of philosophy at Harvard University.

Fact check:  See page 246 of C. I. Lewis: The Last Great Pragmatist ,
by Murray G. Murphey (SUNY Press, 2005).

Figure (a) above is not unrelated to philosophy. See Plato 's Meno  dialogue.
See also a different diamond — a symbol devised by C. I. Lewis for use in
modal logic — in the post Wittgenstein's Diamond (July 10, 2011).

Monday, July 17, 2017

Athens Meets Jerusalem . . .

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

At the Googleplex .

For those whose only interest in higher mathematics
is as a path to the occult

Plato's Diamond and the Hebrew letter Aleph —

          

and some related (if only graphically) mathematics —

Click the above image for some related purely mathematical  remarks.

Monday, June 19, 2017

Final Club

Filed under: General — m759 @ 11:20 am

Today’s New York Times  on a character in a 1978 film —

“Cluelessly upbeat and charmingly idiotic.”

Related material from a post Saturday —

Plato's Formula: A Hollywood version of Plato's diamond from the Meno dialogue

Director with Oscar

Coda —

See as well this  journal on the above date — Sept. 24, 2015.

Thursday, June 15, 2017

Early Personal Computer

Filed under: General — Tags: — m759 @ 10:01 am

(The title is from yesterday morning's Graphical Interfaces.)

Plato's diamond in Jowett's version of the Meno dialogue

Monday, May 15, 2017

Appropriation at MoMA

Filed under: General,Geometry — m759 @ 1:14 pm

For example, Plato's diamond as an object to be transformed —

Plato's diamond in Jowett's version of the Meno dialogue

Versions of the transformed object —

See also The 4×4 Relativity Problem in this journal.

Tuesday, September 27, 2016

Chomsky and Lévi-Strauss in China

Filed under: General,Geometry — Tags: , , — m759 @ 7:31 am

Or:  Philosophy for Jews

From a New Yorker  weblog post dated Dec. 6, 2012 —

"Happy Birthday, Noam Chomsky" by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).

Consider that not-particularly-deep structure from the Meno dialogue
in the light of the following…

The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the Lévi-Strauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji  denotes the fundamental concept in
Chinese philosophy that such a going-beyond is both useful
and possible.)

The matrix at left below represents the feminine yin  principle
and the diamond at right represents the masculine yang .

      From a post of Sept. 22,
"Binary Opposition Illustrated" —

A symbol of the unity of yin and yang —

Related material:

A much more sophisticated approach to the "deep structure" of the
Meno diagram —

The larger cases —

The diamond theorem

Saturday, August 6, 2016

Mystic Correspondence:

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

The Cube and the Hexagram

The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic 
hexagram, or Star of David, with the six cube faces 
mapped to the six hexagram lines and the twelve  
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large). 

Exercise:  Is this noteworthy mapping** of faces to lines, 
edges to points, and vertices to triangles an isolated 
phenomenon, or can it be viewed in a larger context?

* See the discussion at magic-squares.net of
   "perimeter-magic cubes"

** Apparently derived from the Cube + Hexagon figure
    discussed here in various earlier posts. See also
    "Diamonds and Whirls," a note from 1984.

Sunday, June 19, 2016

Making Gatsby Great Again

Filed under: General,Geometry — m759 @ 2:24 pm

Image-- From the Diamond in Plato's Meno to Modern Finite Geometry

See also the previous post.

Tuesday, April 26, 2016

Interacting

Filed under: General,Geometry — Tags: — m759 @ 8:31 pm

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

Monday, April 25, 2016

Seven Seals

Filed under: General,Geometry — Tags: , — m759 @ 11:00 pm

 An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —

"More poetically "

From Hermann Weyl's 1952 classic Symmetry

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

The seven seals from the previous post, with some context —

These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .

Thursday, May 7, 2015

Paradigm for Pedagogues

Filed under: General,Geometry — Tags: , — m759 @ 7:14 pm

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

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

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

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

Tuesday, September 16, 2014

Where the Joints Are

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

An image related to the recent posts Sense and Sensibility:

A quote from yesterday's post The Eight:

A possible source for the above phrase about phenomena "carved at their joints":

See also the carving at the joints of Plato's diamond from the Meno :

Image-- Plato's diamond and a modern version from finite geometry

Related material: Phaedrus on Kant as a diamond cutter
in Zen and the Art of Motorcycle Maintenance .

Wednesday, August 27, 2014

Altar

Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 am

"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"

— Macaulay, quoted in the April 2013 film "Oblivion"

"Leave a space." — Tom Stoppard, "Jumpers"

Related material: The August 16, 2014, sudden death in Scotland
of an architect of the above Cardross seminary, and a Log24 post,
Plato's Logos, from the date of the above photo: June 26, 2010.

See also…

IMAGE- T. Lux Feininger on 'Gestaltung'

Here “eidolon” should instead be “eidos .”

An example of eidos — Plato's diamond (from the Meno ) —

http://www.log24.com/log/pix10A/100607-PlatoDiamond.gif

Sunday, March 2, 2014

Sermon

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

Raiders of the Lost  (Continued)

"Socrates: They say that the soul of man is immortal…."

From August 16, 2012

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not ,  at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

A more correct version, from hermes-press.com —

Socrates: He only guesses that because the square is double, the line is double.Meno: True.

 

Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line?

[Boy] Yes.

Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)?

[Boy] Certainly.

Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet?

[Boy] Yes.

Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet?

[Boy] Yes.

Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN)

[Boy] True.

Socrates: And is not that four times four?

[Boy] Certainly.

Socrates: And four times is not double?

[Boy] No, indeed.

Socrates: But how much?

[Boy] Four times as much.

Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much.

[Boy] True.

Socrates: Four times four are sixteen— are they not?

[Boy] Yes.

As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect  version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:

A 4×4 square version, by Coxeter in 1950, of  a tesseract

This square version we may call the Galois  tesseract.

Tuesday, September 3, 2013

“The Stone” Today Suggests…

Filed under: General,Geometry — m759 @ 12:31 pm

A girl's best friend?

The Philosopher's Gaze , by David Michael Levin,
U. of California Press, 1999, in III.5, "The Field of Vision," pp. 174-175—

The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released  from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as  ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate  respect in the receptivity of a perception that lets itself  be appropriated by  the ground and accordingly lets  the phenomenon of the ground be  what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see  the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt  is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil  underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this  division, decision, and scission, an ekstatic écart  underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged.

* PE: Parmenides  of Heidegger in English— Bloomington: Indiana University Press, 1992

** PG: Parmenides  of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992

Examples of "the primordial Riß " as ἀρχή  —

For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß  as ἀρχή , see
Function Decomposition Over a Finite Field.

Sunday, July 28, 2013

Sermon

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

(Simplicity continued)

"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by  the
demonstration or the metaphor.

What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already-
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."

— Jan Zwicky, Wisdom & Metaphor , page 36 (left)

Zwicky illustrates this with Plato's diamond figure
​from the Meno  on the facing page— her page 36 (right).

A more sophisticated geometrical figure—

Galois-geometry key to
Desargues' theorem:

   D   E   F
 S'  P Q R
 S  P' Q' R'
 O  P1 Q1 R1

For an explanation, see 
Classical Geometry in Light of Galois Geometry.

Tuesday, July 9, 2013

Vril Chick

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

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

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

Compare to an image of Vril muse Maria Orsitsch.

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

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

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

(See also the original catalog page.)

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

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

Thursday, January 31, 2013

Scholarship in 1961…

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

Before Derrida's writings on Plato and on inscription

A remark by the late William Harris:

"Scholarship has many dark ages, and they do not all fall
in the safe confines of remote antiquity."

For more about Harris, see the previous post.

Discussing an approach to solving a geometrical problem 
from section 86e of the Meno , Harris wrote that

"… this is a very important element of method and purpose,
one which must be taken with great seriousness and respect.
In fact it is as good an example of the master describing for us
his method as Plato ever gives us. Tricked by the appearance
of brevity and unwilling to follow through Plato's thought on
the road to Euclid, we have garbled or passed over a unique
piece of philosophical information."

Harris, though not a geometer, was an admirable man.
His remark on the Meno  method is itself worthy of respect.

In memory of Harris, Plato, and pre-Derrida scholarship, here
are some pages from 1961 on the problem Harris discussed.

A pair of figures from the 1961 pages indicates how one view of the
section 86e problem (at right below) resembles the better-known 
demonstration earlier in the Meno  of how to construct
a square of area 2 —

Thursday, December 6, 2012

The Embedding

Filed under: General — m759 @ 6:29 pm

Part I

Embedding the Stone (March 23, 2012) —

The Meno Embedding

Plato's Diamond embedded in The Matrix

Part II

ReverbNation.com — Lawrence Class —

Thursday, August 16, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 8:00 pm

(Continued from August 13. See also Coxeter Graveyard.)

Coxeter exhuming Geometry

Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.

A similar 1950 figure by Coxeter illustrating a tesseract

For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.

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:

Plato's Diamond embedded in The Matrix

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.

Friday, March 23, 2012

Embedding the Stone

Filed under: General,Geometry — Tags: — m759 @ 8:00 am

"Imbedding the God character in a holy book's very detailed narrative
and building an entire culture around this narrative
seems by itself to confer a kind of existence on Him."

John Allen Paulos in the philosophy column "The Stone,"
     New York Times  online, Oct. 24, 2010

A related post from Log24 later that year—

Sunday, November 28, 2010

The Embedding

 — m759 @ 6:00 AM

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

Plato's Diamond embedded in The Matrix

See also Plato's Code and
 Plato Thanks the Academy.

 

 

"Next come the crown of thorns and Jesus' agonized crawl across the stage,
bearing the weight of his own crucifix. And at last, after making
yet another entrance, Mr. Nolan strikes the pose immortalized
in centuries of art, clad in a demure loincloth, arms held out to his sides,
one leg artfully bent in front of the other, head hanging down
in tortured exhaustion. Gently spotlighted, he rises from the stage
as if by magic, while a giant cross, pulsing with hot gold lights,
descends from above to meet him. Mr. Lloyd Webber's churning guitar rock
hits a climactic note, and the audience erupts in excited applause."

— Charles Isherwood, review of "Jesus Christ Superstar" in today's  New York Times

Other remarks on embedding —

Part I

Review of a new book on linguistics, embedding, and a South American tribe—

"Imagine a linguist from Mars lands on Earth to survey the planet's languages…."
Chronicle of Higher Education , March 20, 2012

Part II

The Embedding , by Ian Watson (Review of a 1973 novel from Shakespeare's birthday, 2006)

Thursday, October 20, 2011

The Thing Itself

Filed under: General,Geometry — m759 @ 11:29 am

Suggested by an Oct. 18 piece in the Book Bench section
of the online New Yorker  magazine—

http://www.log24.com/log/pix11C/111020-Derrida.GIF

http://www.log24.com/log/pix11C/111020-Topia122.GIF

Related material suggested by the "Shouts and Murmurs" piece
in The New Yorker , issue dated Oct. 24, 2011—

"a series of e-mails from a preschool teacher planning to celebrate
the Day of the Dead instead of Halloween…"

A search for Coxeter + Graveyard in this journal yields…

Coxeter exhuming Geometry

Here the tombstone says "GEOMETRY… 600 BC — 1900 AD… R.I.P."

A related search for Plato + Tombstone yields an image from July 6, 2007…

The image “http://www.log24.com/log/pix06A/061019-Tombstones.jpg” cannot be displayed, because it contains errors.

Here Plato's poems to Aster suggested
the "Star and Diamond" tombstone.

The eight-rayed star is an ancient symbol of Venus
and the diamond is from Plato's Meno .

The star and diamond are combined in a figure from
12 AM on September 6th, 2011—

The Diamond Star

http://www.log24.com/log/pix11B/110905-StellaOctangulaView.jpg

See Configurations and Squares.

That webpage explains how Coxeter
united the diamond and the star.

Those who prefer narrative to mathematics may consult
a definition of the Spanish word lucero  from March 28, 2003.

Thursday, August 4, 2011

Midnight in Oslo

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

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

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

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

Plato's Diamond

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

Paradigms

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

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

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

Plato's paradigm in the Meno

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

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

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

Aspects of the paradigm change—

Monochrome figures to
   colored figures

Areas to
   transformations

Continuous transformations to
   non-continuous transformations

Euclidean geometry to
   finite geometry

Euclidean quantities to
   finite fields

The 24 patterns resulting from the paradigm change—

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

Each pattern has some ordinary or color-interchange symmetry.

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

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

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

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

"Remember me to her."

— Closing words of the Algis Budrys novel Rogue Moon .

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

Sunday, April 10, 2011

Bedeviled

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

From tonight's online New York Times

John McCracken, Sculptor of Geometric Forms, Dies at 76

McCracken died in Manhattan on Friday, April 8.

From Christopher Knight in tonight's online LA Times

… the works embody perceptual and philosophical conundrums. The colored planks stand on the floor like sculptures….

McCracken was bedeviled by Stanley Kubrick's famously obscure science-fiction epic, "2001: A Space Odyssey," with its iconic image of an ancient monolith floating in outer space. The 1968 blockbuster was released two years after the artist made his first plank.

"At the time, some people thought I had designed the monolith or that it had been derived from my work," he told art critic Frances Colpitt of the coincidence in a 1998 interview.

Two photos of McCracken's 1967 Black Plank  seem relevant—

November 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCrackenPlank1967400w.jpg

December 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCracken-NatGallery-NothingToSeeHere-400w.jpg

Material that an artist might view as related, if only synchronistically—

Two posts in this journal on the dates the photos were taken—
The Embedding on November 28 and Dry Bones on December 28.

The photos are of an exhibition titled "There is nothing to see here" at the
National Gallery of Art, October 30, 2010-April 24, 2011 —

Click to enlarge.

http://www.log24.com/log/pix11/110410-NothingToSee-400w.jpg

For related nihilism from the National Gallery, see "Pictures of Nothing" in this journal.

Some less nihilistic illustrations—

The Meno  Embedding

Plato's Diamond embedded in The Matrix

A photo by one of the artists whose work is displayed above beside McCracken's—

http://www.log24.com/log/pix11/110410-Sugimoto-AndoChurch.jpg

"Accentuate the Positive."
 — Clint Eastwood

Thursday, February 17, 2011

Paradigms

Filed under: General,Geometry — Tags: , — m759 @ 4:16 pm

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

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

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

Plato's paradigm in the Meno

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

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

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

Aspects of the paradigm change* —

Monochrome figures to
colored figures

Areas to
transformations

Continuous transformations to
non-continuous transformations

Euclidean geometry to
finite geometry

Euclidean quantities to
finite fields

Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.

* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα  above, along with the commentators' specific reference to
the Meno  figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.

Sunday, November 28, 2010

The Embedding

Filed under: General,Geometry — m759 @ 6:00 am

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

Plato's Diamond embedded in The Matrix

See also Plato's Code and
 Plato Thanks the Academy.

Monday, October 4, 2010

Stone Junction

Filed under: General — m759 @ 6:29 am

Continued from May 18, 2010.

Previous logo for the New York Times  feature "The Stone"—

http://www.log24.com/log/pix10B/101004-IntroducingTheStone100518.jpg

Today's new logo, appearing retroactively

http://www.log24.com/log/pix10B/101004-NYT-newstonelogo.png

Comparison—

http://www.log24.com/log/pix10A/100518-TheStoneNYT.jpg   http://www.log24.com/log/pix10B/101004-NYT-thestone75.gif

From the October 3 "The Stone," Hegel on Wall Street

The “Phenomenology” is a philosophical portrait gallery that presents depictions, one after another, of different, fundamental ways in which individuals and societies have understood themselves.  Each self-understanding has two parts: an account of how a particular kind of self understands itself and, then, an account of the world that the self considers its natural counterpart.  Hegel narrates how each formation of self and world collapses because of a mismatch between self-conception and how that self conceives of the larger world.  Hegel thinks we can see how history has been driven by misshapen forms of life in which the self-understanding of agents and the worldly practices they participate in fail to correspond.  With great drama, he claims that his narrative is a “highway of despair.”

J.M. Bernstein of the New School for Social Research

A two-part self-understanding that is not  from Hegel—

1. An account of how a particular kind of self understands itself:

            … world’s wildfire, leave but ash:
                In a flash, at a trumpet crash,
I am all at once what Christ is, ' since he was what I am, and
This Jack, joke, poor potsherd, ' patch, matchwood, immortal diamond,
                Is immortal diamond.

2. An account of the world that the self considers its natural counterpart:

CLOUD-PUFFBALL, torn tufts, tossed pillows ' flaunt forth, then chevy on an air-
built thoroughfare: heaven-roysterers, in gay-gangs ' they throng; they glitter in marches.
Down roughcast, down dazzling whitewash, ' wherever an elm arches,
Shivelights and shadowtackle in long ' lashes lace, lance, and pair.
Delightfully the bright wind boisterous ' ropes, wrestles, beats earth bare
Of yestertempest’s creases; in pool and rut peel parches
Squandering ooze to squeezed ' dough, crust, dust; stanches, starches
Squadroned masks and manmarks ' treadmire toil there
Footfretted in it. Million-fuelèd, ' nature’s bonfire burns on.

Sunday, October 3, 2010

Search for the Basic Picture

Filed under: General,Geometry — Tags: , — m759 @ 5:01 pm

(Click to enlarge.)

http://www.log24.com/log/pix10B/101003-SambinBasicPictureSearch.jpg

The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."

That search was suggested by the title of today's New York Times  op-ed essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.

Further information on one of the images above—

http://www.log24.com/log/pix10B/101003-VisualThinkingSm.jpg

A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics  yields the following—

Thursday July 5, 2007

m759 @ 7:11 PM

In defense of Plato’s realism

(vs. sophists’ nominalism– see recent entries.)

Plato cited geometry, notably in the Meno , in defense of his realism.
Consideration of the Meno 's diamond figure leads to the following:

The Eightfold Cube and its Inner Structure

For the Meno 's diamond figure in Giaquinto, see a review—

http://www.log24.com/log/pix10B/101003-VisualThinkingReview.jpg

— Review by Jeremy Avigad (preprint)

Finite geometry supplies a rather different context for Plato's  "basic picture."

In that context, the Klein four-group often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)

The Times  op-ed essay today notes that linguistic  translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."

The list of four-group transformations in the mathematical  sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.

Thursday, July 1, 2010

Plato’s Code

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

John Allen Paulos yesterday at Twitter

"Plato's code cracked? http://bit.ly/ad6k1S
Fascinating if not a hoax or hype."

The story that Paulos linked to is about a British
academic who claims to have found some
symbolism hidden in Plato's writings by
splitting each into 12 parts and correlating
the 12 parts with semitones of a musical scale.

I prefer a different approach to Plato that is
related to the following hoax and hype—

HOAX:

From Dan Brown's novel Angels & Demons  (2000)

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

HYPE:

Image-- From 'Alchemy,' by Holmyard, the diamond of Aristotle's 4 elements and 4 qualities

This  four-elements diamond summarizes the classical
four elements and four qualities neatly, but some scholars
might call the figure "hype" since it deals with an academically
disreputable subject, alchemy, and since its origin is unclear.

For the four elements' role in some literature more respectable
than Dan Brown's, see Poetry's Bones.

Although an author like Brown might spin the remarks
below into a narrative—  The Plato Code — they are
neither  hoax nor hype.

NOT  HOAX:

Image-- From the Diamond in Plato's Meno to Modern Finite Geometry

NOT  HYPE:

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

For related non-hoax, non-hype remarks, see
The Rational Enterprise: Logos in Plato's Theaetetus,
by Rosemary Desjardins.

Those who prefer  hoax and hype in their philosophy may consult
the writings of, say, Barbara Johnson, Rosalind Krauss, and—
in yesterday's NY Times's  "The Stone" columnNancy Bauer.

Image-- The Philosophers' Stone according to The New York Times

— The New York Times

Saturday, June 26, 2010

Plato’s Logos

Filed under: General,Geometry — Tags: , — 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

Monday, June 7, 2010

Inspirational Combinatorics

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

According to the Mathematical Association of America this morning, one purpose of the upcoming June/July issue of the Notices of the American Mathematical Society  is

"…to stress the inspirational role of combinatorics…."

Here is another contribution along those lines—

Eidetic Variation

from page 244 of
From Combinatorics to Philosophy: The Legacy of  G.-C. Rota,
hardcover, published by Springer on August 4, 2009

(Edited by Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, and Fabrizio Palombi)

"Rota's Philosophical Insights," by Massimo Mugnai—

"… In other words, 'objectivism' is the attitude [that tries] to render a particular aspect absolute and dominant over the others; it is a kind of narrow-mindedness attempting to reduce to only one the multiple layers which constitute what we call 'reality.' According to Rota, this narrow-mindedness limits in an essential way even of [sic ] the most basic facts of our cognitive activity, as, for example, the understanding of a simple declarative sentence: 'So objectivism is the error we [make when we] persist in believing that we can understand what a declarative sentence means without a possible thematization of this declarative sentence in one of [an] endless variety of possible contexts' (Rota, 1991*, p. 155). Rota here implicitly refers to what, amongst phenomenologists is known as eidetic variation, i.e. the change of perspective, imposed by experience or performed voluntarily, from which to look at things, facts or sentences of the world. A typical example, proposed by Heidegger, in Sein und Zeit  (1927) and repeated many times by Rota, is that of the hammer."

* Rota, G.-C. (1991), The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT, Cambridge, MA, MIT Mathematics Department

The example of the hammer appears also on yesterday's online New York Times  front page—

http://www.log24.com/log/pix10A/100606-Touchstones.jpg

Related material:

From The Blackwell Dictionary of Western Philosophy

Eidetic variation — an alternative expression for eidetic reduction

Eidetic reduction

Husserl's term for an intuitive act toward an essence or universal, in contrast to an empirical intuition or perception. He also called this act an essential intuition, eidetic intuition, or eidetic variation. In Greek, eideo  means “to see” and what is seen is an eidos  (Platonic Form), that is, the common characteristic of a number of entities or regularities in experience. For Plato, eidos  means what is seen by the eye of the soul and is identical with essence. Husserl also called this act “ideation,” for ideo  is synonymous with eideo  and also means “to see” in Greek. Correspondingly, idea  is identical to eidos.

An example of eidos— Plato's diamond (from the Meno )—

http://www.log24.com/log/pix10A/100607-PlatoDiamond.gif

For examples of variation of this eidos, see the diamond theorem.
See also Blockheads (8/22/08).

Related poetic remarks— The Trials of Device.

Monday, November 10, 2008

Monday November 10, 2008

Filed under: General,Geometry — Tags: — m759 @ 10:31 am

Frame Tales

From June 30

("Will this be on the test?")

Frame Tale One:

Summer Reading

The King and the Corpse: Tales of the Soul's Conquest of Evil, by Heinrich Zimmer

Subtitle:
Tales of the Soul's
Conquest of Evil

Frame Tale Two:

Barry Sharples
on his version of the
  Kaleidoscope Puzzle

Background:

"A possible origin of this puzzle is found in a dialogue
 between Socrates and Meno written by the Greek philosopher,
 Plato, where a square is drawn inside
a square such that
the blue square is twice the area  of the yellow square.

Plato's Diamond

Colouring the triangles produces a starting pattern
which is a one-diamond figure made up of four tiles
and there are 24 different possible arrangements."

Twenty-four Variations on a Theme of Plato

The King and the Corpse  —

"The king asked, in compensation for his toils during this strangest
of all the nights he had ever known, that the twenty-four riddle tales
told him by the specter, together with the story of the night itself,
should be made known over the whole earth
and remain eternally famous among men."

Frame Tale Three:

Finnegans Wake

"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."

Wednesday, October 8, 2008

Wednesday October 8, 2008

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

Serious Numbers

A Yom Kippur
Meditation

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution

Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–

"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"

(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)

Richard J. Trudeau on the 'Story Theory' of truth

Excerpt from
The Non-Euclidean Revolution

What does this have to do with numbers?

Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.

Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.

For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.

Saturday, June 28, 2008

Saturday June 28, 2008

Filed under: General — Tags: — m759 @ 12:00 pm
The God Factor

NY Lottery June 23, 2008: Mid-day 322, Evening 000


The following poem of Emily Dickinson is quoted here in memory of John Watson Foster Dulles, a scholar of Brazilian history who died at 95 on June 23.  He was the eldest son of Secretary of State John Foster Dulles, a nephew of Director of Central Intelligence Allen Dulles, brother of Roman Catholic Cardinal Avery Dulles, and a grandson of Presbyterian minister Allen Macy Dulles, author of The True Church.

I asked no other thing,   
No other was denied.   
I offered Being for it;   
The mighty merchant smiled.   
 
Brazil? He twirled a button,           
Without a glance my way:   
"But, madam, is there nothing else   
That we can show to-day?"


"He twirled a button…."

Plato's diamond figure from the 'Meno'

The above figure
of Plato
(see 3/22)
was suggested by
Lacan's diamond
Lacan's lozenge - said by some to symbolize Derrida's 'differance'
(losange or poinçon)
as a symbol —
according to Frida Saal
of Derrida's
différance
which is, in turn,
"that which enables and
results from Being itself"
—  according to
Professor John Lye

I prefer Plato and Dulles
to Lacan and Lye.
 

Sunday, April 27, 2008

Sunday April 27, 2008

Filed under: General,Geometry — m759 @ 8:28 am
Happy Birthday
 
to the late
Gian-Carlo Rota,
mathematician and
scholar of philosophy

Rota* on his favorite philosopher:

“I believe Husserl to be the greatest philosopher of all times….

Intellectual honesty is the striking quality of Husserl’s writings. He wrote what he honestly believed to be true, neither more nor less. However, honesty is not clarity; as a matter of fact, honesty and clarity are at opposite ends. Husserl proudly refused to stoop to the demands of showmanship that are indispensable in effective communication.”

B.C. by Hart, April 27, 2008:  Discovery of the Wheel and of the Diamond

Related material:
 
The Diamond Theorem

 

* Gian-Carlo Rota, “Ten Remarks on Husserl and Phenomenology,” in O.K. Wiegand et al. (eds.), Phenomenology on Kant, German Idealism, Hermeneutics and Logic, pp. 89-97, Kluwer Academic Publishers, 2000

Wednesday, July 25, 2007

Wednesday July 25, 2007

Filed under: General,Geometry — Tags: , — m759 @ 9:00 am
The Comedy of
George Tabori

George Tabori

From AP “Obituaries in the News”–
Filed with The New York Times
at 11:16 p.m. ET July 24, 2007–

George Tabori

“BERLIN (AP) — Hungarian-born playwright and director George Tabori, a legend in Germany’s postwar theater world whose avant-garde works confronted anti-Semitism, died Monday [July 23, 2007]. He was 93.

Tabori, who as recently as three years ago dreamed of returning to stage to play the title role in Shakespeare’s ‘King Lear,’ died in his apartment near the theater, the Berliner Ensemble said Tuesday, noting that friends and family had accompanied him through his final days. No cause of death was given.

Born into a Jewish family in Budapest on May 24, 1914, Tabori fled in 1936 to London, where he started working for the British Broadcasting Corp., and became a British citizen. His father, and other members of his family, were killed at Auschwitz.

Tabori moved to Hollywood in the 1950s, where he worked as a scriptwriter, most notably co-writing the script for Alfred Hitchcock’s 1953 film, ‘I Confess.’

He moved to Germany in the 1970s and launched a theater career that spanned from acting to directing to writing. He used sharp wit and humor in his plays to examine the relationship between Germany and the Jews, as well as attack anti-Semitism.

Among his best-known works are ‘Mein Kampf,’ set in the Viennese hostel where Adolf Hitler lived from 1910-1913, and the ‘Goldberg Variations,’ both dark farces that poke fun at the Nazis.”

From Year of Jewish Culture:

“The year 2006 marks the 100th anniversary of the establishment of the Jewish Museum in Prague.”

From the related page Programme (October-December):

Divadlo v Dlouhé
George Tabori: GOLDBERGOVSKÉ VARIACE / THE GOLDBERG VARIATIONS, 19 October, 7 p.m. A comedy on creation and martyrdom.”

Variations on
Birth and Death

From Log24 on the date of
the Prague production of the
Tabori “Goldberg Variations,”
an illustration in honor of
Sir Thomas Browne, who
was born, and died,
on that date:

Laves tiling

The above is from
Variable Resolution 4–k Meshes:
Concepts and Applications
(pdf),
by Luiz Velho and Jonas Gomes.

See also Symmetry Framed
and The Garden of Cyrus.

Variations on
the Afterlife

 From Log24
on the date of
Tabori’s death:

Theme

(Plato, Meno)

Plato's Diamond colored

and Variations:

Diamond Theory cover, 1976

Click on “variations” above
for some material on
the “Goldberg Variations”
of Johann Sebastian Bach.

 

Monday, July 23, 2007

Monday July 23, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:59 am
 
Today’s Birthday:
Daniel Radcliffe
(“Harry Potter”)

Harry Potter and the Philosopher's Stone DVD

Theme

(Plato, Meno)

Plato's Diamond colored

and Variations:

Diamond Theory cover, 1976
Click on picture for details.

“A diamond jubilance
beyond the fire,
That gives its power
to the wild-ringed eye”

— Wallace Stevens,
“The Owl in the Sarcophagus”

Thursday, July 5, 2007

Thursday July 5, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:11 pm

In Defense of
Plato’s Realism

(vs. sophists’ nominalism–
see recent entries.)

Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:

The Eightfold Cube and its Inner Structure

Click on image for details.

As noted in an entry,
Plato, Pegasus, and
the Evening Star,

linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:

Trigram of K'un, the Receptive

K’un
The Receptive

_____________________________________________

Update of Nov. 30, 2013:

From  a little-known website in Kuala Lumpur:
(Click to enlarge.)

The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General,Geometry — m759 @ 9:00 pm
Logos and Logic
(private, cut from prev. entry)

The diamond is used in modal logic to symbolize possibility.

  The 3×3 grid may also be used
to illustrate “possibility.”  It leads,
as noted at finitegeometry.org, to
the famed “24-cell,” which may be
pictured either as the diamond
figure from Plato’s Meno

The image “http://www.log24.com/theory/images/poly-24cell-sm.jpg” cannot be displayed, because it contains errors.

Click for details.

  — or as a figure
with 24 vertices:

The image “http://www.log24.com/theory/images/poly-24cell-02sm.jpg” cannot be displayed, because it contains errors.

Click for details.

The “diamond” version of the
24-cell seems unrelated to the
second version that shows all
vertices and edges, yet the
second version is implicit,
or hidden, in the first.
Hence “possibility.”

Neither version of the 24-cell
seems related in any obvious
way to the 3×3 grid, yet both
versions are implicit,
or hidden, in the grid.
Hence “possibility.”

Tuesday, October 3, 2006

Tuesday October 3, 2006

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

Serious

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

Charles Matthews at Wikipedia, Oct. 2, 2006

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

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

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

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

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

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

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

Tuesday, July 11, 2006

Tuesday July 11, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:11 pm

Not Crazy Enough?

Some children of the sixties may feel that today's previous two entries, on Syd Barrett, the Crazy Diamond, are not crazy enough.  Let them consult the times of those entries– 2:11 and 8:15– and interpret those times, crazily, as dates: 2/11 and 8/15.

This brings us to Stephen King territory– apparently the natural habitat of Syd Barrett.

See Log24 on a 2/11, Along Came a Dreamcatcher, and Log24 on an 8/15, The Line.

From 8/15, a remark of Plato:

"There appears to be a sort of war of Giants and Gods going on…"

(Compare with the remarks by Abraham Cowley for Tom Stoppard's recent birthday.)

From 2/11, two links: Halloween Meditations  and We Are the Key.

From Dreamcatcher (the film and the book):

The image “http://www.log24.com/log/pix06/060211-Freeman2.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06/060324-Dreamcatcher.gif” cannot be displayed, because it contains errors.

For Syd Barrett as Duddits,

The image “http://www.log24.com/log/pix06A/060711-Duddits.jpg” cannot be displayed, because it contains errors.

see Terry Kirby on Syd Barrett
(edited– as in Stephen King
and the New Testament
for narrative effect):

"He appeared as the Floyd performed the song 'Shine On You Crazy Diamond.' It contains the words: 'Remember when you were young, you shone like the sun. Shine on you crazy diamond. Now there's a look in your eyes, like black holes in the sky.'

At first, they didn't recognise the man, whose head and eyebrows were shaved….

But this was the 'crazy diamond' himself: Syd Barrett, the subject of the song….

When Roger Waters saw his old friend, he broke down….

Rick Wright, the keyboards player, later told an interviewer:

… 'Roger [Waters] was in tears, I think I was; we were both in tears. It was very shocking… seven years of no contact and then to walk in while we're actually doing that particular track. I don't know – coincidence, karma, fate, who knows? But it was very, very, very powerful.'"

Remarks suitable for Duddits's opponent, Mister Gray, may be found in the 1994 Ph.D. thesis of Noel Gray.

"I refer here to Plato's utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave."

Plato's Diamond

Shine on, gentle Duddits.

Thursday, December 8, 2005

Thursday December 8, 2005

Filed under: General,Geometry — Tags: , — m759 @ 2:56 pm
Aion Flux
 
That Nature is a Heraclitean Fire…
— Poem title, Gerard Manley Hopkins  

From Jung's Map of the Soul, by Murray Stein:

"… Jung thinks of the self as undergoing continual transformation during the course of a lifetime…. At the end of his late work Aion, Jung presents a diagram to illustrate the dynamic movements of the self…."

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

"The formula presents a symbol of the self, for the self is not just a stable quantity or constant form, but is also a dynamic process.  In the same way, the ancients saw the imago Dei in man not as a mere imprint, as a sort of lifeless, stereotyped impression, but as an active force…. The four transformations represent a process of restoration or rejuvenation taking place, as it were, inside the self…."

"The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity. 

What the formula can only hint at, however, is the higher plane that is reached through the process of transformation and integration. The 'sublimation' or progress or qualitative change consists in an unfolding of totality into four parts four times, which means nothing less than its becoming conscious. When psychic contents are split up into four aspects, it means that they have been subjected to discrimination by the four orienting functions of consciousness. Only the production of these four aspects makes a total description possible. The process depicted by our formula changes the originally unconscious totality into a conscious one." 

— Jung, Collected Works, Vol. 9, Part 2, Aion: Researches into the Phenomenology of the Self (1951) 

Related material: 

  The diamond theorem

"Although 'wholeness' seems at first sight to be nothing but an abstract idea (like anima and animus), it is nevertheless empirical in so far as it is anticipated by the psyche in the form of  spontaneous or autonomous symbols. These are the quaternity or mandala symbols, which occur not only in the dreams of modern people who have never heard of them, but are widely disseminated in the historical recods of many peoples and many epochs. Their significance as symbols of unity and totality is amply confirmed by history as well as by empirical psychology.  What at first looks like an abstract idea stands in reality for something that exists and can be experienced, that demonstrates its a priori presence spontaneously. Wholeness is thus an objective factor that confronts the subject independently of him… Unity and totality stand at the highest point on the scale of objective values because their symbols can no longer be distinguished from the imago Dei. Hence all statements about the God-image apply also to the empirical symbols of totality."

— Jung, Aion, as quoted in
Carl Jung and Thomas Merton

Thursday, May 26, 2005

Thursday May 26, 2005

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

Drama of the Diagonal
"The beautiful in mathematics
resides in contradiction.
Incommensurability, logoi alogoi, was
the first splendor in mathematics."
— Simone Weil, Oeuvres Choisies,
éd. Quarto
, Gallimard, 1999, p. 100
 

Logos Alogos
by S. H. Cullinane

"To a mathematician, mathematical entities have their own existence, they habitate spaces created by their intention.  They do things, things happen to them, they relate to one another.  We can imagine on their behalf all sorts of stories, providing they don't contradict what we know of them.  The drama of the diagonal, of the square…"

— Dennis Guedj, abstract of "The Drama of Mathematics," a talk to be given this July at the Mykonos conference on mathematics and narrative.

For the drama of the diagonal of the square, see

Thursday May 26, 2005

Filed under: General,Geometry — m759 @ 4:00 am
The Changing

The previous entry dealt with a transformation
of the diamond figure from Plato’s Meno
into a visual proof of the Pythagorean theorem:

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

Here is a transformation of Plato’s diamond
into the “gyronny” pattern of heraldry:

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

The image “http://www.log24.com/theory/images/DiamondChanging2.jpg” cannot be displayed, because it contains errors.

For the mathematics dealing with
this sort of transformation, see
The Diamond 16 Puzzle and Diamond Theory.

Wednesday, May 25, 2005

Wednesday May 25, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:22 pm

The Turning

Readers who have an Amazon.com account may view book pages relevant to the previous entry.  See page 77 of The Way We Think, by Fauconnier and Turner (Amazon search term = Meno).  This page discusses both the Pythagorean theorem and Plato's diamond figure in the Meno, but fails to "blend" these two topics.  See also page 53 of The History of Mathematics, by Roger Cooke (first edition), where these two topics are in fact blended (Amazon search term = Pythagorean).  The illustration below is drawn from the Cooke book.

The image “http://www.log24.com/log/pix05/050525-Figs.jpg” cannot be displayed, because it contains errors.

Cooke demonstrates how the Pythagorean theorem might have been derived by "blending" Plato's diamond (left) with the idea of moving the diamond's corners (right).

The previous entry dealt with a conference on mathematics and narrative.  Above is an example I like of mathematics…. Here is an example I like of narrative:

Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn  certain  that
it  was  the  sort of experience that her mother would not have
approved of on a first date.
     "Is this all part of what we have to do to go to  Asgard?"
she said. "Or are you just fooling around?"
     "We will go to Asgard...now," he said.
     At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect  was as if he had twisted the entire world through a
billionth part of a billionth  part  of  a  degree.  Everything
shifted,  was  for  a  moment  minutely  out of focus, and then
snapped back again as a suddenly different world.

— Douglas Adams, The Long Dark Tea-Time of the Soul

And here is a blend of the concepts "Asgard" and "conference":

"Asgard
    During the Interuniverse Society conference,
    a bridge was opened to Valhalla…."

  Bifrost
     In Norse myth, the rainbow bridge
     that connected Earth to Asgard,
     home of the gods.  It was extended
     to Tellus Tertius during the
     Interuniverse Society conference"

— From A Heinlein Concordance

The image “http://www.log24.com/log/pix05/050525-Rainbow.jpg” cannot be displayed, because it contains errors.

 

— Front page picture from a
local morning newspaper published
today, Wednesday, May 25, 2005

 

As George Balanchine once asked,
"How much story do you want?"

Sunday, May 22, 2005

Sunday May 22, 2005

Filed under: General — Tags: — m759 @ 4:09 pm

The Diamond in the Labyrinth

From the labyrinth of Solitude:

  1. An Invariant Feast
  2. Columbia News obituary of Robert D. Cumming
  3. Solitude

    (Phenomenology and Deconstruction, Vol 4
    by Robert Denoon Cumming)

    On page 13 of Solitude —

    From Heidegger’s “Letter on Humanism” —
    “… so long as philosophy merely busies itself with continually obstructing the possibility of admission to the subject of thinking– that is, the truth of being– it escapes the danger of ever being broken against the hardness of that subject.  Thus ‘philosophizing’ about the shattering is separated by an abyss from a thinking that is shattered.”

    This suggests a search for
    “Heidegger” + “diamond,” which yields —

  4. The Diamond at the End of Time,
    which leads to
  5. Orson Welles Interviews Jilly Dybka,
    which leads to
  6. Poetry Hut Blog,
    which leads to

  7. Fair Territory, by Jilly Dybka,
    which contains the following —
  8. The Quickening

    I hold my breath, the plane’s
       wheels under me
    still suspended in the minutes after
    takeoff, when the planet’s brute gravity
    statistically can cause a disaster.
    We are flying low enough that I scan
    civilization in miniature.
    Blue pill swimming pools, and
       roadways that fan
    out like ribbons in the wind. On the sure
    crust, too, a baseball diamond.
       Young boys race
    across the tilted surface, mute and small,
    kicking up red dust. First base, Second base,
    Third Base, Home. We ascend into nightfall
    and beneath the broken stars one kid bunts.
    I remember I was a rookie once.

  9. From yesterday’s online New York Times:

    The image “http://www.log24.com/log/pix05/050521-ConeyIslandCrash.gif” cannot be displayed, because it contains errors.

Nine is a Vine.

Tuesday, March 22, 2005

Tuesday March 22, 2005

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

Make a Différance

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

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

 


From a Contemporary Literary Theory website:

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

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

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

 

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



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

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

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

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


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

Thursday, January 27, 2005

Thursday January 27, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:29 am
Crystal Night

From artbook.com:

Mies van der Rohe:
Mies in Berlin

The image “http://www.log24.com/log/pix05/050127-Mies.jpg” cannot be displayed, because it contains errors.

Winner of
The Society of Architectural Historians
2002 Philip Johnson Award
for Excellence

Exhibition Catalog

"Published to accompany
a groundbreaking 2001 exhibition at
The Museum of Modern Art, New York."

 

From Mies and the Mastodon,
by Martin Filler, The New Republic,
issue dated Aug. 6, 2001:

"It would have been wiser for the new MoMA catalog… to have addressed the issue of his politics…. By ignoring such a central subject… the show gives off a mild stench of cover-up…. Only the German-born Rosemarie Haag Bletter (full disclosure: she is my wife) alludes to the verboten topic in her [catalog] essay on Mies's flirtation with crystal imagery, drawing a sharp parallel between the architect's extensive use of Kristallglas (plate glass) and the ensuing devastation of Kristallnacht, which erupted just three months after he left for the States."

Also from Filler's essay:

"Mies's rigorously simplified structures, typified by grids of steel and glass and an absence of applied ornament, represented the Platonic ideal of modernism for many people."

For more on history, politics, and
Mies's disciple Philip Johnson,
who died Tuesday evening, see

"We Cannot Not Know History."

For more on aesthetics, see the
Log24.net entry of Tuesday noon,

Diamonds Are Forever.

For more on a Platonic ideal of sorts,
see the following figure in two versions:
 
Version A, from Plato's Meno and
Diamond Theory,

The image “http://www.log24.com/log/pix05/050127-MenoDiamond.gif” cannot be displayed, because it contains errors.

and Version B,

The image “http://www.log24.com/log/pix05/050125-Forever.gif” cannot be displayed, because it contains errors.

from the date of Johnson's death
at his "famous crystalline box."

Was less more?

Sunday, August 8, 2004

Sunday August 8, 2004

Filed under: General — Tags: , , — m759 @ 6:08 am

Quartet

An illustration from July 26,
Jung’s birthday and the date
of Alexander Hammid’s death:

 

Jung’s Model
of the Self:


Four Quartets:

“… history is a pattern
Of timeless moments.”


Gerard Malanga, 2003

Alexander
Hammid

From today’s
New York Times
:

Alexander Hammid,
96, Filmmaker
Known for Many Styles,
Dies

By KATHRYN L. SHATTUCK

Published: August 8, 2004

Alexander Hammid, a filmmaker whose body of work spanned the genesis of the experimental movement in Czechoslovakia, early anti-Nazi documentaries and soaring modern Imax spectacles, died on July 26 at his home in Manhattan. He was 96.

His work in the 1950’s and early 60’s involved his passion for the arts,  [including] … a collaboration on Gian Carlo Menotti’s opera “The Medium” and a documentary series of master classes by the cellist Pablo Casals….

 

“… legend has it, supported by Casals himself, that he was conceived when Brahms began his B-flat Major Quartet, of which Casals owned the original manuscript, and that he was born when Brahms completed its composition.”

http://www.bach-cantatas.com/
Bio/Casals-Pablo.htm

Sunday, March 7, 2004

Sunday March 7, 2004

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

Apartments

From Wallace Stevens,
"Notes Toward a Supreme Fiction":

It is the celestial ennui of apartments
That sends us back to the first idea, the quick
Of this invention; and yet so poisonous

Are the ravishments of truth, so fatal to
The truth itself, the first idea becomes
The hermit in a poet’s metaphors,

Who comes and goes and comes and goes all day.
May there be an ennui of the first idea?
What else, prodigious scholar, should there be?….

From Guyan Robertson,
Groups Acting on Affine Buildings
and their Boundaries
:

From Plato's Meno:

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

See Logos and Logic
and the previous entry.

Thursday, February 19, 2004

Thursday February 19, 2004

Filed under: General — m759 @ 9:22 pm

What is Poetry, Part II —

Gombrich vs. Gadamer

Excerpts from
Tetsuhiro Kato on

Gombrich and the
Hermeneutics of Art

Kato on Gombrich

“… according to Gombrich, an image is susceptible to become a target for ‘symbol detectives’…. But the hidden authorial intention… ([for example]… astrology, recalling the famous warning of Panofsky [1955: 32]), almost always tends to become a reproduction of the interpreter’s own ideological prejudice. Not to give into the irrationalism such psychological overinterpretation might invite…. we have to look for the origin of meaning… in…  the social context…. The event of image making is not the faithful transcription of the outside world by an innocent eye, but it is the result of the artist’s act of selecting the ‘nearest equivalence’… based on social convention….”

Kato on Gadamer 

“For [Gadamer], picture reading is a process where a beholder encounters a picture as addressing him or her with a kind of personal question, and the understanding develops in the form of its answer (Gadamer 1981: 23-24; Gadamer 1985: 97,102-103).  But, it must be noted that by this Gadamer does not mean to identify the understanding of an image with some sort of ‘subsumption’ of the image into its meaning (Gadamer 1985: 100). He insists rather that we can understand an image only by actualizing what is implied in the work, and engage in a dialogue with it. This process is ideally repeated again and again, and implies different relations than the original conditions that gave birth to the work in the beginning (Gadamer 1985: 100).

What matters here for Gadamer is to let the aesthetic aspect of image take its own ‘Zeitgestalt’ (Gadamer 1985: 101).”

Example (?) — the Zeitgestalt
of today’s previous entry:

See, too,
 The Quality of Diamond.

Kato’s References:

Gadamer, Hans-Georg. 1981. “Philosophie und Literatur: Was ist die Literatur?,” Phänomenologische Forschungen 11 (1981): 18-45.

Gadamer, Hans-Georg. 1985. “Über das Lesen von Bauten und Bildern.” Modernität und Tradition: Festschrift für Max Imdahl zum 60. Geburtstag. Ed. Gottfried Boehm, Karlheinz Stierle, Gundorf Winter. Munchen: Wilhelm Fink. 97-103.

Panofsky, Erwin. 1955. Meaning in the Visual Arts: Papers in and on Art History. New York: Anchor.

Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: , — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Further Reading:

Monday, March 24, 2003

Monday March 24, 2003

Filed under: General — m759 @ 12:52 pm

Orwell’s question, according to
an admirer of leftist Noam Chomsky:

“When so much of the BS is right out in the open,
why is it that we know so little about it?
Why don’t we see what’s right in front of our eyes?”


Oscar
Deep Chomsky:
Lying, Truth-Telling,
and the Social Order
 
 
 
 
 Michael
 Moore

“First of all, I’d like to thank the Academy….”
— Quotation attributed to Plato

The New Yorker of March 31, 2003, discusses leftist academic Noam Chomsky.  The online edition provides a web page listing pro-Chomsky links.

Chomsky’s influence is based in part on the popularity of his half-baked theories on linguistics, starting in the 1950’s with “deep structure” and “transformational,” or “generative,” grammar.

Chomsky has abandoned many of his previous ideas and currently touts what he calls The Minimalist Program.

For some background on Chomsky’s recent linguistic notions, see the expository essay “Syntactic Theory,” by Elly van Gelderen of the Arizona State University English Department.  Van Gelderen lists her leftist political agenda on her “Other Interests” page.  Her department may serve as an example of how leftists have converted many English departments in American universities to propaganda factories.

Some attacks on Chomsky’s scholarship:

The Emperor’s New Linguistics

The New Grammarians’ Funeral

Beyond Chomsky

Could Chomsky Be Wrong? 

Forty-four Reasons Why the Chomskians Are Mistaken

Call for Papers, Chomsky 2003

Chomsky’s (Mis)Understanding of Human Thinking

Anatomy of a Revolution… Chomsky in 1962

…Linguistic Theory: The Rationality of Noam Chomsky

A Bibliography

Some attacks on Chomsky’s propaganda:

LeftWatch.com Chomsky page

Destructive Generation excerpt

The Sick Mind of Noam Chomsky

Partners in Hate: Noam Chomsky and the Holocaust Deniers

Chomsky and Plato’s Diamond

Like another purveyor of leftist nonsense, Jacques Derrida, Chomsky is fond of citing Plato as a precedent.  In particular, what Chomsky calls “Plato’s problem” is discussed in Plato’s Meno.  For a look at the diamond figure that plays a central role in that dialogue, see Diamond Theory.  For an excellent overview of related material in Plato, see Theory of Forms.

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General — Tags: — m759 @ 4:44 pm

ART WARS:

From The New Yorker, issue of March 17, 2003, Clive James on Aldous Huxley:

The Perennial Philosophy, his 1945 book compounding all the positive thoughts of West and East into a tutti-frutti of moral uplift, was the equivalent, in its day, of It Takes a Village: there was nothing in it to object to, and that, of course, was the objection.”

For a cultural artifact that is less questionably perennial, see Huxley’s story “Young Archimedes.”

Plato, Pythagoras, and
the diamond figure

Plato’s Diamond in the Meno
Plato as a precursor of Gerard Manley Hopkins’s “immortal diamond.” An illustration shows the ur-diamond figure.

Plato’s Diamond Revisited
Ivars Peterson’s Nov. 27, 2000 column “Square of the Hypotenuse” which discusses the diamond figure as used by Pythagoras (perhaps) and Plato. Other references to the use of Plato’s diamond in the proof of the Pythagorean theorem:

Huxley:

“… and he proceeded to prove the theorem of Pythagoras — not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself….
‘You see,’ he said, ‘it seemed to me so beautiful….’
I nodded. ‘Yes, it’s very beautiful,’ I said — ‘it’s very beautiful indeed.'”
— Aldous Huxley, “Young Archimedes,” in Collected Short Stories, Harper, 1957, pp. 246 – 247

Heath:

Sir Thomas L. Heath, in his commentary on Euclid I.47, asks how Pythagoreans discovered the Pythagorean theorem and the irrationality of the diagonal of a unit square. His answer? Plato’s diamond.
(See Heath, Sir Thomas Little (1861-1940),
The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publications, New York, 1956.

Other sites on the alleged
“diamond” proof of Pythagoras

Colorful diagrams at Cut-the-Knot

Illustrated legend of the diamond proof

Babylonian version of the diamond proof

For further details of Huxley’s story, see

The Practice of Mathematics,

Part I, by Robert P. Langlands, from a lecture series at the Institute for Advanced Study, Princeton.

From the New Yorker Contributors page for St. Patrick’s Day, 2003:

Clive James (Books, p. 143) has a new collection, As of This Writing: The Essential Essays, 1968-2002, which will be published in June.”

See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.

Wednesday, November 27, 2002

Wednesday November 27, 2002

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

Waiting for Logos

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

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

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

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

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

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

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

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

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

— Simone Weil

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

— Tom Hinkle, review of Waiting for God

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

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

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

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

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

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

Weil concludes the preceding essay with the following passage:

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

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

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

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

— Blaise Pascal, The Art of Persuasion

 

For more diamond metaphors and Jungian analysis, see

The Diamond Archetype.

Thursday, October 31, 2002

Thursday October 31, 2002

Filed under: General,Geometry — m759 @ 11:07 pm

Plato's
Diamond

From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:

"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…" 

Charles H. Bennett has written about Omega as a cabalistic number.

Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).

From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:

"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."

From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:

SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….

From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:

Galileo on Plato's diamond:

"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"

Roger Bacon on Plato's diamond:

"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"

It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.

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