Log24

Saturday, April 4, 2020

Devising Entities…

Filed under: General — m759 @ 9:34 am

Continues.

The previous post  displayed a photo from November 2014.

Remarks quoted here  in November 2014

“Before time began, there was the Cube.”
— Optimus Prime

Tuesday, December 20, 2016

Enumerating Entities

Filed under: General,Geometry — m759 @ 7:29 pm

See also a Terry Gilliam film on "crunching entities."

Saturday, October 10, 2015

Nonphysical Entities

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

Norwegian Sculpture Biennial 2015 catalog, p. 70 —

" 'Ambassadørene' er fysiske former som presenterer
ikk-fysiske fenomener. "

Translation by Google —

" 'Ambassadors' physical forms presents
nonphysical phenomena. "

Related definition —

Are the "line diagrams" of the diamond theorem and
the analogous "plane diagrams" of the eightfold cube
nonphysical entities? Discuss.

Friday, July 3, 2015

Crunching Entities*

Filed under: General — m759 @ 9:19 pm

A figure I prefer to the "Golden Tablet" of Night at the Museum —

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

The source — The Log24 post "Zero System" of July 31, 2014.

* For the title, see The New Yorker  of Sept. 22, 2014.

Saturday, September 20, 2014

The Metaphysics of Entities

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

Anthony Lane in The New Yorker , issue dated Sept. 22, 2014:

"The hero of 'The Zero Theorem' is a computer genius called Qohen Leth
(Christoph Waltz)…. He is the sole resident of a derelict church, where,
on a crucifix in front of the altar, the head of Christ has been replaced by
a security camera. No prayers are ever said, and none are answered.

In short, the place is deconsecrated, but to claim that it lacks any spark of
sacred yearning would be wrong, because Qohen devotes his days to seeking
the Zero Theorem, which—whatever it may be—lies at the fuzzy limit of
human powers. We crunch entities,” he says, as if that explained anything.
His employer is Mancom, a large corporation that, in Orwellian fashion,
oversees ordinary lives, although it betrays more frantic desperation than
glowering threat."

One approach to the metaphysics of entities was indicated in the previous
post, 'Metaphysics for Gilliam." A different approach:

"Categories, Sets, and the Nature of Mathematical Entities,"
by Jean-Pierre Marquis, Ch. 13, pp. 181-192, in the 2006 book
The Age of Alternative Logics , ed. by van Benthem et al.
(Springer, Netherlands).

From pages 182-183 —

13.2 The nature of mathematical entities

Let us start with the nature of mathematical entities in general and with a
rough and classical distinction that will simply set the stage for the picture we
want to develop. We essentially follow Lowe 1998* for the basic distinctions. We
need to distinguish between abstract and concrete entities, on the one hand, and
universals and particulars on the other hand. For our purpose, it is not necessary
to specify a criterion of demarcation between abstract and concrete entities. We
simply assume that such a distinction can be made, e.g. concrete entities can
change whereas abstract entities cannot. We assume that a universal is an entity
that can be instantiated by entities which themselves are not instantiable, the
latter being of course particulars. Given these distinctions, an entity can be a
concrete particular, a concrete universal, an abstract particular or an abstract
universal.

Our focus here is between the last two possibilities. For we claim that the
current conception of sets makes them abstract particulars whereas for objects
defined within categories, mathematical entities are abstract universals. This,
we claim, is true of category theory as it is.

* Lowe, E.J., 1998, The Possibility of Metaphysics , Oxford: Clarendon Press.

Tuesday, February 5, 2013

Entities

Filed under: General — Tags: — m759 @ 12:24 pm

From January 26, 2013

IMAGE- Yahoo CEO Marissa Mayer at Davos and the ontology of entities

Related material: "universe of discourse"

Saturday, July 3, 2010

Devising Entities

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

or, Darkness and Brightness at Noon

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional."

— Don DeLillo, Point Omega

GF(4) = {0, 1, ω, ω2}

Symbolic representation of a Galois field

"One two three  four,
  who are we  for?"

— Cheerleaders' chant

Monday, February 6, 2023

Interality Studies

Filed under: General — Tags: , — m759 @ 12:26 pm
 

You, Xi-lin; Zhang, Peter. "Interality in Heidegger." 
The Free Library , April 1, 2015.  
. . . .

The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi.
. . . .

Some remarks from this  journal on April 1, 2015 —

Manifest O

Tags:  

— m759 @ 4:44 AM April 1, 2015

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral  group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21 An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01 Portrait of O  A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16 Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15 Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."

Wednesday, August 24, 2022

Metaphor

Filed under: General — m759 @ 2:53 pm

Saturday, January 16, 2021

Dark City . . .

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

Continues .

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

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

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

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

A different “shorter way” —

Wednesday, January 13, 2021

Solid

Filed under: General — m759 @ 3:26 pm

“None of the memories and identities of the people in the city are solid.”

— Grimstrup, Jesper Møller.  SHELL BEACH:
The search for
the final theory .  Kindle Edition.

See also Peter Woit’s  “Various Links” post  today and Shell Beach  in this  journal.

For another approach to solidity (from pure  mathematics, not physics), see
a Log24 search for Solid Symmetry.

Saturday, January 9, 2021

Kierkegaard Revisited

Filed under: General — m759 @ 9:15 pm

Kierkegaard is imagining a society where
people’s identities are robbed
under the disguise of modern liberation.”

As in . . .

http://www.log24.com/log/pix18/180903-Womens_Night_Bingo-at48.41-The_Net.jpg

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

Tuesday, July 21, 2020

Oeuvre

Filed under: General — m759 @ 5:13 am
From “Nabokov’s Crosswords of Composition,” by
Rebecca Freeh-Maciorowski, a paper presented at NEMLA, dated 15 October 2014 —

“In a way, Nabokov’s entire oeuvre might be built upon one all-encompassing ‘crossword,’ a possibility raised by W.W. Rowe when he writes ‘Words and phrases seem faintly but undeniably to catch many others in the prism of their associations and connotations, almost as if Nabokov’s entire oeuvre were planned from the very start’ (viii). Turning to Pale Fire , the work of Simon Rowberry provides evidence of a whole network of ‘themed entries’ within this novel, what Rowberry refers to as ‘the novel’s promiscuous intertextuality.’ Alternately, the points and coordinates that Nabokov refers to constitute the composition’s ‘checked cells.’ The checked cells are the basic mechanism of the crossword puzzle; essentially, they are the guiding force of the entire puzzle, controlling both the construction and solution. These are the cells within the crossword puzzle in which two words intersect. In Nabokov’s compositional crossword, the ‘checked cells’ are those points which combine disparate entities, places of intersection, where objects and themes converge.”

Rowe, W.W., Nabokov’s Deceptive World , New York University Press, 1971.

Rowberry, Simon, “Pale Fire  as a Hypertextual Network.” 22nd ACM Hypertext Conf., Eindhoven, Netherlands. 6-9 June 2011. Web.

The Rowberry date appears to be, specifically, 8  June 2011:

A Kinbote note — See also this  journal on 8 June 2011.

Update of 3:03 PM ET the same day —

In keeping with Kinbote’s character as an unreliable narrator . . .
Rowberry’s Eindhoven slides  indicate he spoke on 9  June 2011.

See as well the Log24 post  “Historical Fiction” from June 2011.

Friday, September 27, 2019

The Black List

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

"… Max Black, the Cornell philosopher, and others have pointed out
how 'perhaps every science must start with metaphor and end with
algebra, and perhaps without the metaphor there would never have
been any algebra' …."

— Max Black, Models and Metaphors, Cornell U. Press, 1962,
page 242, as quoted in Dramas, Fields, and Metaphors, by 
Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
 

Metaphor —

Algebra —

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985.

Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964.

Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982.

Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

— Wallace Stevens, "The Motive for Metaphor"

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 11:29 pm

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Saturday, September 16, 2017

The Zero Monstrance

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

From "The Metaphysics of Entities," a post of Sept. 20, 2014 —

Anthony Lane in The New Yorker  on a 2013 film —

"The hero of 'The Zero Theorem' is a computer genius
called Qohen Leth (Christoph Waltz)…. He is the sole
resident of a derelict church, where, on a crucifix in front
of the altar, the head of Christ has been replaced by a
security camera. No prayers are ever said, and none are
answered."

Related dialogue from a 2008 film

Another view of the Zero Theorem derelict church —

Thursday, May 11, 2017

In Memoriam

Filed under: General — m759 @ 9:00 pm

See also Chandrasekharan in a Log24 search for Weyl+Schema.

Update of 6:16 AM Friday, May 12, 2017 —

The phrase "smallest perfect universe" is from Burkard Polster (2001).

Wednesday, March 8, 2017

Inscapes

Filed under: General,Geometry — Tags: — m759 @ 6:42 pm

"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

Illustration

Commentary

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985.

Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964.

Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982.

Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Monday, September 19, 2016

Squaring the Pentagon

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

The "points" and "lines" of finite  geometry are abstract
entities satisfying only whatever incidence requirements
yield non-contradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, non-Euclidean)
space.

Models  of finite geometries may, however, embed the
points and lines within non -finite geometries in order
to aid visualization.

For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not*  lines of the Euclidean plane.

Example — See "4×4" in this journal.

Some impose on configurations from finite geometry
the rather artificial requirement that both  points and lines
must be representable as those of a Euclidean plane.

Example:  A Cremona-Richmond pentagon —

Pentagon with pentagram

A square version of these 15 "points" —

A 1905 square version of these 15 "points" 
with digits instead of letters —

See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).

Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2-subsets of a 6-set." (See one such search.)

* But in some models are subsets of the grid lines 
   that separate squares within an array.

Friday, November 20, 2015

Alyssa’s Maxim

Filed under: General — m759 @ 12:00 pm

Thursday, October 16, 2014

Taking the Fork

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

Related material:  Alyssa Milano in this journal —

IMAGE- Alyssa Milano as a child, with fork

Tuesday, September 23, 2014

Both Hands and an Ass Map

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

Remarks by University Diaries  this morning suggested a search for
Rutgers in this journal. That search yields a post from Dec. 30, 2005,
that is closely related to both this morning's previous post and to recent
Log24 remarks on entities and concrete universals .

Sunday, September 21, 2014

Saturday-Morning Concept

Filed under: General,Geometry — m759 @ 7:59 pm

Why Is Our Sci-Fi So Glum About A.I.?,”
by Jayson Greene, NY Times Sunday Magazine  today —

“You come to pity these advanced beings, bumping against
the dunderheaded constraints that their less-advanced
creators have placed on them. Johansson’s Lucy grows so
powerful as her cerebral capacity multiplies that she is able
to manipulate her cellular structure. And yet, when pursued
by an entire planet’s worth of law enforcement, she settles
on a disguise straight out of Saturday-morning cartoons —
really big sunglasses and a hairdo change.”

See also this  journal on Saturday morning for a definition, and
Geometry of the I Ching for examples, of

changeable, instantiable entities, i.e., concrete universals.

Above the entrance to Plato's Academy: AGEOMETRETOS MEDEIS EISITO

Saturday, September 20, 2014

Ennead Boo

Filed under: General — m759 @ 12:00 pm

A search in this journal for Yahoo Entities ended with a link to another
Log24 search, Nine Years, which in turn suggested…

A scene from the current TV series “Intruders
(Season 1, Episode 1, at 9:22 of 45 min.)

Friday, September 19, 2014

Metaphysics for Gilliam

Filed under: General — m759 @ 9:29 pm

See also…

Related remarks: Diederik Aerts at arXiv.org.

See also Aerts (as above) on the metaphysics of entities  (1984):

Tuesday, January 14, 2014

Release Date

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

The premiere of the Lily Collins film Abduction 
(see previous post) was reportedly in Sydney, Australia,
on August 23, 2011.

From that date in this journal

IMAGE- The eight Galois quaternions

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

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

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

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

— Fritz Leiber, short story, 1959

Related material from Wikipedia, suggested by the reference quoted
in this morning's post to "a four-dimensionalist (perdurantist) ontology"—

"… perdurantism also applies if one believes there are temporal
but non-spatial abstract entities (like immaterial souls…)."

Wednesday, September 4, 2013

Secretary

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

Margalit Fox in The New York Times  this evening

Judith Daniels, the founding editor in chief of Savvy ,
the first glossy magazine aimed at executive women,
died on Sunday at her home in Union, Me. She was 74….

Savvy  will not tell you how to be a good secretary,”
one of its early promotional fliers read. “Savvy  will tell you
how to hire a good secretary— and how to fire.”

From the date of Daniels' death:  The Crossword Omen.

See, too,  Vogue  in this journal and Ontology.

From a post of January 26, 2013

IMAGE- Yahoo CEO Marissa Mayer and the ontology of entities

Sally in the 2013 film Oblivion : "Are you an effective team?"

Tuesday, June 18, 2013

Mise-en-Scène

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

IMAGE- 'Lexicon,' a novel by Max Barry published June 18, 2013

This journal on May 14, 2013:

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."

— Paul Valéry, Introduction to the Method of
    Leonardo da Vinci

But first, a word from our sponsor

Tuesday, May 14, 2013

Commercial

Filed under: General,Geometry — m759 @ 7:00 am

(Continued from December 30, 2012)

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."

— Paul Valéry, Introduction to the Method of
    Leonardo da Vinci

But first, a word from our sponsor

Brought to you by two uploads, each from Sept. 11, 2012—

Symmetry and Hierarchy and the above VINCI Genius commercial.

Sunday, January 27, 2013

Shining Forth

Filed under: General — m759 @ 12:09 am

See noon yesterday 

IMAGE- Yahoo's Marissa Mayer on the ontology of entities

and the date of Donald Hornig’s death:

Saturday, January 26, 2013

Nine Years

Filed under: General — m759 @ 12:00 pm

IMAGE- Yahoo CEO Marissa Mayer and the ontology of entities

Excerpt from an essay cached nine years ago:

"The current dominant conceptual framework
which pictures the self as an inner entity
is slowly breaking up. And I am convinced that
some, if not all, of the approaches to the self
sketched here will form the basis for a new
conceptual framework…."

Context for the essay: 

A journal issue titled "The Opening of Narrative Space" (pdf, 475 KB)

For one sort of narrative space, see Giordano Bruno in this journal.

See also Nine Years.

Friday, December 28, 2012

Cube Koan

Filed under: General,Geometry — Tags: , , , , — m759 @ 4:56 am
 

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Monday, October 8, 2012

Issue 16

Filed under: General,Geometry — Tags: , , — m759 @ 8:14 pm

From triplecanopy, Issue 16 —

International Art English, by Alix Rule and David Levine (July 30, 2012)

… In what follows, we examine some of the curious lexical, grammatical, and stylistic features of what we call International Art English. We consider IAE’s origins, and speculate about the future of this language through which contemporary art is created, promoted, sold, and understood. Some will read our argument as an overelaborate joke. But there’s nothing funny about this language to its users. And the scale of its use testifies to the stakes involved. We are quite serious….*

Space  is an especially important word in IAE and can refer to a raft of entities not traditionally thought of as spatial (the space of humanity ) as well as ones that are in most circumstances quite obviously spatial (the space of the gallery ). An announcement for the 2010 exhibition “Jimmie Durham and His Metonymic Banquet,” at Proyecto de Arte Contemporáneo Murcia in Spain, had the artist “questioning the division between inside and outside in the Western sacred space”—the venue was a former church—“to highlight what is excluded in order to invest the sanctum with its spatial purity. Pieces of cement, wire, refrigerators, barrels, bits of glass and residues of ‘the sacred,’ speak of the space of the exhibition hall … transforming it into a kind of ‘temple of confusion.’”

Spatial and nonspatial space are interchangeable in IAE. The critic John Kelsey, for instance, writes that artist Rachel Harrison “causes an immediate confusion between the space of retail and the space of subjective construction.” The rules for space  in this regard also apply to field , as in “the field of the real”—which is where, according to art historian Carrie Lambert-Beatty, “the parafictional has one foot.” (Prefixes like para -, proto -, post -, and hyper – expand the lexicon exponentially and Germanly, which is to say without adding any new words.) It’s not just that IAE is rife with spacey terms like intersection , parallel , parallelism , void , enfold , involution , and platform …

* Footnote not in the original—
  See also Geometry and Death from the date of the above article.

Saturday, January 7, 2012

Past Tense

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

From a post that was written for Twelfth Night

Bernhard Weiss on the philosophy of Michael Dummett—

" … debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent
of the way we represent that aspect to ourselves. For example,
is there a realm of mathematical entities that exists fully formed
independently of our mathematical activity? Are there facts about
the past that our use of the past tense aims to capture?"

Yes and Yes.

See also The Whirligig of Time in this journal.

Thursday, January 5, 2012

Precisely

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

From a review of Truth and Other Enigmas , a book by the late Michael Dummett—

"… two issues stand out as central, recurring as they do in many of the
essays. One issue is the set of debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent of the way we
represent that aspect to ourselves. For example, is there a realm of mathematical
entities that exists fully formed independently of our mathematical activity? Are
there facts about the past that our use of the past tense aims to capture? The other
issue is the view
which Dummett learns primarily from the later Wittgenstein
that the meaning of an expression is fully determined by its use, by the way it
is employed by speakers. Much of his work consists in attempts to argue for this
thesis, to clarify its content and to work out its consequences. For Dummett one
of the most important consequences of the thesis concerns the realism debate and
for many other philosophers the prime importance of his work precisely consists
in this perception of a link between these two issues."

Bernhard Weiss, pp. 104-125 in Central Works of Philosophy , Vol. 5,
ed. by John Shand,
McGill-Queen's University Press, June 12, 2006

The above publication date (June 12, 2006) suggests a review of other
philosophical remarks related to that date. See …

http://www.log24.com/log/pix12/120105-SpekkensExcerpt.jpg

For some more-personal remarks on Dummett, see yesterday afternoon's
"The Stone" weblog in The New York Times.

I caught the sudden look of some dead master….

Four Quartets

Saturday, October 1, 2011

Like an Orb

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

It turns out that Fabrizio Palombi, author and editor of books on the late combinatorialist-philosopher Gian-Carlo Rota, is also an expert on the French charlatan Lacan. (For recent remarks related to Rota, see yesterday's Primordiality and the link "6.7 (June 7)" in today's The Crowe Sphere.)

"We all have our little mythologies."

— "Lacan’s Mathematics," by Amadou Guissé, Alexandre Leupin, and Steven D. Wallace (a preprint from the website of Steven D. Wallace, assistant professor of mathematics at Macon State College, Macon, GA.) A more extensive quote from "Lacan's Mathematics"—

Epistemological Cuts* or Births?

An epistemological cut can be described as the production of homonyms. For example, the word orb in Ptolemaic cosmology and the same word in the Kepler’s system, albeit similar, designate two entities that have nothing in common: the first one, in the Ancients’ cosmology, is a crystal sphere to which stars are attached; orb, for Kepler, is an ellipsis whose sole material existence is the algorithm describing its path. A cut becomes major when all word of different eras change meaning. A case in point is the cut between polytheism and monotheism (Judaism): the word god or god takes an entirely different meaning, and this change affects all areas of a vision of the world. From the non created world of the Ancients, inhabited by eternal Gods, we pass on to a world created by a unique God, who is outside of his creation. This cut affects all areas of thinking. However, mythology, albeit separated from the new vision by the cut, survives as an enduring residue. Our sexual thinking, for example, is essential mythological, as proven by the endurance of the Oedipus complex or our cult of this ancient deity called Eros. Love is inherently tied to what Freud called the omnipotence of thought or magical thinking.

Of course, the quintessential major epistemological cut for us is the break effectuated by modern science in the 17th century. All the names are affected by it: however, who can claim he or she has been entirely purged of pre-scientific reasoning? Despite us living in a scientific universe, we all have our little mythologies, residues of an era before the major epistemological cut.

Any modeling of major epistemological cuts, or paradigm changes as Thomas Kuhn would have it, has therefore to account at the same time for a complete break with past names (that is, new visions of the world) as well as the survival of old names and mythologies.

* For some background on this Marxist jargon, see Epistemological Break (La Coupure Épistémologique ) at the website Concept and Form: The Cahiers pour  l’Analyse  and Contemporary French Thought.

Friday, July 16, 2010

Point Omega continued

Filed under: General — m759 @ 1:26 pm

"We tried to create new realities overnight…."
Point Omega, quoted here in the post
Devising Entities (July 3, 2010)

Image-- NY Lottery, evening  July 15=000, midday July 16=911

See also last night's Meditation as well as the earlier posts
Language Game and The Subject Par Excellence.

Saturday, July 3, 2010

Beyond the Limits

Filed under: General,Geometry — Tags: , , — m759 @ 7:29 pm

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…."

– Don DeLillo, Point Omega

Capitalized, the letter omega figures in the theology of two Jesuits, Teilhard de Chardin and Gerard Manley Hopkins. For the former, see a review of DeLillo. For the latter, see James Finn Cotter's Inscape  and "Hopkins and Augustine."

The lower-case omega is found in the standard symbolic representation of the Galois field GF(4)—

GF(4) = {0, 1, ω, ω2}

A representation of GF(4) that goes beyond the standard representation—

http://www.log24.com/log/pix10A/100703-Elements.gif

Here the four diagonally-divided two-color squares represent the four elements of GF(4).

The graphic properties of these design elements are closely related to the algebraic properties of GF(4).

This is demonstrated by a decomposition theorem used in the proof of the diamond theorem.

To what extent these theorems are part of "a saga of created reality" may be debated.

I prefer the Platonist's "discovered, not created" side of the debate.

Monday, June 28, 2010

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 pm

See David Corfield,
"The Robustness of Mathematical Entities."

This is an abstract from a paper at a conference,
"From Practice to Results in Logic and Mathematics"
(June 21st-23rd, 2010, Archives Henri Poincaré,
University of Nancy (France)).

See also Corfield's post "Inevitability in Mathematics"
at the n-Category Café today. He links to an earlier
post, "Mathematical Robustness." From that post—

…let’s see what Michiel Hazewinkel has to say
 in his paper Niceness theorems:

It appears that many important mathematical objects
(including counterexamples) are unreasonably
nice, beautiful and elegant. They tend to have
(many) more (nice) properties and extra bits
of structure than one would a priori expect….

Wednesday, June 23, 2010

Group Theory and Philosophy

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

Excerpts from "The Concept of Group and the Theory of Perception,"
by Ernst Cassirer, Philosophy and Phenomenological Research,
Volume V, Number 1, September, 1944.
(Published in French in the Journal de Psychologie, 1938, pp. 368-414.)

The group-theoretical interpretation of the fundaments of geometry is,
from the standpoint of pure logic, of great importance, since it enables us to
state the problem of the "universality" of mathematical concepts in simple
and precise form and thus to disentangle it from the difficulties and ambigui-
ties with which it is beset in its usual formulation. Since the times of the
great controversies about the status of universals in the Middle Ages, logic
and psychology have always been troubled with these ambiguities….

Our foregoing reflections on the concept of group  permit us to define more
precisely what is involved in, and meant by, that "rule" which renders both
geometrical and perceptual concepts universal. The rule may, in simple
and exact terms, be defined as that group of transformations  with regard to
which the variation of the particular image is considered. We have seen
above that this conception operates as the constitutive principle in the con-
struction of the universe of mathematical concepts….

                                                              …Within Euclidean geometry,
a "triangle" is conceived of as a pure geometrical "essence," and this
essence is regarded as invariant with respect to that "principal group" of
spatial transformations to which Euclidean geometry refers, viz., displace-
ments, transformations by similarity. But it must always be possible to
exhibit any particular figure, chosen from this infinite class, as a concrete
and intuitively representable object. Greek mathematics could not
dispense with this requirement which is rooted in a fundamental principle
of Greek philosophy, the principle of the correlatedness of "logos" and
"eidos." It is, however, characteristic of the modern development of
mathematics, that this bond between "logos" and "eidos," which was indis-
soluble for Greek thought, has been loosened more and more, to be, in the
end, completely broken….

                                                            …This process has come to its logical
conclusion and systematic completion in the development of modern group-
theory. Geometrical figures  are no longer regarded as fundamental, as
date of perception or immediate intuition. The "nature" or "essence" of a
figure is defined in terms of the operations  which may be said to
generate the figure.
The operations in question are, in turn, subject to
certain group conditions….

                                                                                                    …What we
find in both cases are invariances with respect to variations undergone by
the primitive elements out of which a form is constructed. The peculiar
kind of "identity" that is attributed to apparently altogether heterogen-
eous figures in virtue of their being transformable into one another by means
of certain operations defining a group, is thus seen to exist also in the
domain of perception. This identity permits us not only to single out ele-
ments but also to grasp "structures" in perception. To the mathematical
concept of "transformability" there corresponds, in the domain of per-
ception, the concept of "transposability." The theory  of the latter con-
cept has been worked out step by step and its development has gone through
various stages….
                                                                                 …By the acceptance of
"form" as a primitive concept, psychological theory has freed it from the
character of contingency  which it possessed for its first founders. The inter-
pretation of perception as a mere mosaic of sensations, a "bundle" of simple
sense-impressions has proved untenable…. 

                             …In the domain of mathematics this state of affairs mani-
fests itself in the impossibility of searching for invariant properties of a
figure except with reference to a group. As long as there existed but one
form of geometry, i.e., as long as Euclidean geometry was considered as the
geometry kat' exochen  this fact was somehow concealed. It was possible
to assume implicitly  the principal group of spatial transformations that lies
at the basis of Euclidean geometry. With the advent of non-Euclidean
geometries, however, it became indispensable to have a complete and sys-
tematic survey of the different "geometries," i.e., the different theories of
invariancy that result from the choice of certain groups of transformation.
This is the task which F. Klein set to himself and which he brought to a
certain logical fulfillment in his Vergleichende Untersuchungen ueber neuere
geometrische Forschungen
….

                                                          …Without discrimination between the
accidental and the substantial, the transitory and the permanent, there
would be no constitution of an objective reality.

This process, unceasingly operative in perception and, so to speak, ex-
pressing the inner dynamics of the latter, seems to have come to final per-
fection, when we go beyond perception to enter into the domain of pure
thought. For the logical advantage and peculiar privilege of the pure con –
cept seems to consist in the replacement of fluctuating perception by some-
thing precise and exactly determined. The pure concept does not lose
itself in the flux of appearances; it tends from "becoming" toward "being,"
from dynamics toward statics. In this achievement philosophers have
ever seen the genuine meaning and value of geometry. When Plato re-
gards geometry as the prerequisite to philosophical knowledge, it is because
geometry alone renders accessible the realm of things eternal; tou gar aei
ontos he geometrike gnosis estin
. Can there be degrees or levels of objec-
tive knowledge in this realm of eternal being, or does not rather knowledge
attain here an absolute maximum? Ancient geometry cannot but answer
in the affirmative to this question. For ancient geometry, in the classical
form it received from Euclid, there was such a maximum, a non plus ultra.
But modern group theory thinking has brought about a remarkable change
In this matter. Group theory is far from challenging the truth of Euclidean
metrical geometry, but it does challenge its claim to definitiveness. Each
geometry is considered as a theory of invariants of a certain group; the
groups themselves may be classified in the order of increasing generality.
The "principal group" of transformations which underlies Euclidean geome-
try permits us to establish a number of properties that are invariant with
respect to the transformations in question. But when we pass from this
"principal group" to another, by including, for example, affinitive and pro-
jective transformations, all that we had established thus far and which,
from the point of view of Euclidean geometry, looked like a definitive result
and a consolidated achievement, becomes fluctuating again. With every
extension of the principal group, some of the properties that we had taken
for invariant are lost. We come to other properties that may be hierar-
chically arranged. Many differences that are considered as essential
within ordinary metrical geometry, may now prove "accidental." With
reference to the new group-principle they appear as "unessential" modifica-
tions….

                 … From the point of view of modern geometrical systematization,
geometrical judgments, however "true" in themselves, are nevertheless not
all of them equally "essential" and necessary. Modern geometry
endeavors to attain progressively to more and more fundamental strata of
spatial determination. The depth of these strata depends upon the com-
prehensiveness of the concept of group; it is proportional to the strictness of
the conditions that must be satisfied by the invariance that is a universal
postulate with respect to geometrical entities. Thus the objective truth
and structure of space cannot be apprehended at a single glance, but have to
be progressively  discovered and established. If geometrical thought is to
achieve this discovery, the conceptual means that it employs must become
more and more universal….

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.

Friday, April 16, 2010

The Craft, continued

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

Image--Movie poster for 'The Craft'

"Honesty's the best policy."
— Miguel de Cervantes   

"Liars prosper."
— Anonymous   

— Epigraphs to On Writing:
A Memoir of the Craft
,
by Stephen King

Lavender Blue,
Dilly, Dilly,
Lavender Green…
Image-- Spacek as Carrie

The cruelest month continues…

"…as Newton conceived it, the distinction between
the individualities of two particles is so marked that
it is impossible for them ever to coincide or for
either of them to alter the being of the other…."

"Waves interfere with each other because they are
interchangeable and thus not distinguishable;
two processes can coincide in space and time
but two substances cannot. Thus the wave
reveals a whole new possibility of identity…."

"The concept of a field is elusive."

— Peter Pesic, Seeing Double: Shared Identities
in Physics, Philosophy, and Literature
,
Chapter 6, "The Fields of Light"

Saturday, December 26, 2009

Annals of Philosophy

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

Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:

"Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries."

For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square.  The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions— well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.

Friday, August 10, 2007

Friday August 10, 2007

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

The Ring of Gyges

10:31:32 AM ET

Commentary by Richard Wilhelm
on I Ching Hexagram 32:

“Duration is… not a state of rest, for mere standstill is regression.
Duration is rather the self-contained and therefore self-renewing
movement of an organized, firmly integrated whole, taking place in
accordance with immutable laws and beginning anew at every ending.”

Related material

The Ring of the Diamond Theorem

Jung and the Imago Dei

Log24 on June 10, 2007: 

WHAT MAKES IAGO EVIL? some people ask. I never ask. —Joan Didion

Iago states that he is not who he is. —Mark F. Frisch

“Not Being There,”
by Christopher Caldwell
,
from next Sunday’s
New York Times Magazine:

“The chance to try on fresh identities was the great boon that life online was supposed to afford us. Multiuser role-playing games and discussion groups would be venues for living out fantasies. Shielded by anonymity, everyone could now pass a ‘second life’ online as Thor the Motorcycle Sex God or the Sage of Wherever. Some warned, though, that there were other possibilities. The Stanford Internet expert Lawrence Lessig likened online anonymity to the ring of invisibility that surrounds the shepherd Gyges in one of Plato’s dialogues. Under such circumstances, Plato feared, no one is ‘of such an iron nature that he would stand fast in justice.’Time, along with a string of sock-puppet scandals, has proved Lessig and Plato right.”

“The Boy Who Lived,”
by Christopher Hitchens
,
from next Sunday’s
New York Times Book Review:

On the conclusion of the Harry Potter series:”The toys have been put firmly back in the box, the wand has been folded up, and the conjuror is discreetly accepting payment while the children clamor for fresh entertainments. (I recommend that they graduate to Philip Pullman, whose daemon scheme is finer than any patronus.)”

I, on the other hand,
recommend Tolkien…
or, for those who are
already familiar with
Tolkien, Plato– to whom
The Ring of Gyges” may
serve as an introduction.

“It’s all in Plato, all in Plato:
bless me, what do they
teach them at these schools!”
C. S. Lewis

Saturday, July 21, 2007

Saturday July 21, 2007

Filed under: General — Tags: — m759 @ 9:45 am

Death of a Nominalist

“All our words from loose using have lost their edge.” –Ernest Hemingway

(The Hemingway quotation is from the AP’s “Today in History” on July 21, 2007; for the context, see Death in the Afternoon.)

Today seems as good a day as any for noting the death of an author previously discussed in Log24 on January 29, 2007, and January 31, 2007.

Joseph Goguen
died on July 3, 2006. (I learned of his death only after the entries of January 2007 were written. They still hold.)

Goguen’s death may be viewed in the context of the ongoing war between the realism of Plato and the nominalism of the sophists. (See, for instance, Log24 on August 10-15, 2004, and on July 3-5, 2007.)

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

“Before introducing algebraic semiotics and structural blending, it is good to be clear about their philosophical orientation. The reason for taking special care with this is that, in Western culture, mathematical formalisms are often given a status beyond what they deserve. For example, Euclid wrote, ‘The laws of nature are but the mathematical thoughts of God.’ Similarly, the ‘situations’ in the situation semantics of Barwise and Perry, which resemble conceptual spaces (but are more sophisticated– perhaps too sophisticated), are considered to be actually existing, real entities [23], even though they may include what are normally considered judgements.5 The classical semiotics of Charles Sanders Peirce [24] also tends towards a Platonist view of signs. The viewpoint of this paper is that all formalisms are constructed in the course of some task, such as scientific study or engineering design, for the heuristic purpose of facilitating consideration of certain issues in that task. Under this view, all theories are situated social entities, mathematical theories no less than others; of course, this does not mean that they are not useful.”

5 The “types” of situation theory are even further removed from concrete reality.

[23] Jon Barwise and John Perry. Situations and Attitudes. MIT (Bradford), 1983.
[24] Charles Sanders Peirce. Collected Papers. Harvard, 1965. In 6 volumes; see especially Volume 2: Elements of Logic.

From Log24 on the date of Goguen’s death:

Requiem for a clown:

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

Norman Mailer

This same Mailer aphorism was quoted, along with an excerpt from the Goguen passage above, in Log24 this year on the date of Norman Mailer’s birth.  Also quoted on that date:

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

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

Whether the above excerpt– from Hans Christian Oersted‘s The Soul in Nature (1852)– is superior to the similar remark of Goguen, the reader may decide.

Thursday, July 12, 2007

Thursday July 12, 2007

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

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

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

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

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

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

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

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

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

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

"Values are rooted
in narrative."

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

Related material:

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

Tuesday, May 8, 2007

Tuesday May 8, 2007

Filed under: General — m759 @ 2:56 pm
The Public Square

Center of Town, Cuernavaca, from Paul Goodman's Communitas

On the words “symbology” and “communitas” (the former used, notably, as the name of a fictional field at Harvard in the novel The Da Vinci Code)–

Symbology:

“Also known as ‘processual symbolic analysis,’ this concept was developed by Victor Turner in the mid-1970s to refer to the use of symbols within cultural contexts, in particular ritual. In anthropology, symbology originated as part of Victor Turner’s concept of ‘comparative symbology.’ Turner (1920-1983) was professor of Anthropology at Cornell University, the University of Chicago, and finally he was Professor of Anthropology and Religion at the University of Virginia.” —Wikipedia

Symbology and Communitas:

 From Beth Barrie’s
  Victor Turner
“‘The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole’ (Turner, 1967:51). Turner considered himself a comparative symbologist, which suggests he valued his contributions to the study of ritual symbols. It is in the closely related study of ritual processes that he had the most impact.

The most important contribution Turner made to the field of anthropology is his work on liminality and communitas. Believing the liminal stage to be of ‘crucial importance’ in the ritual process, Turner explored the idea of liminality more seriously than other anthropologists of his day.

As noted earlier Turner elaborated on van Gennep’s concept of liminality in rites of passage. Liminality is a state of being in between phases. In a rite of passage the individual in the liminal phase is neither a member of the group she previously belonged to nor is she a member of the group she will belong to upon the completion of the rite. The most obvious example is the teenager who is neither an adult nor a child. ‘Liminal entities are neither here nor there; they are betwixt and between the positions assigned and arrayed by law, custom, convention, and ceremonial’ (Turner, 1969:95). Turner extended the liminal concept to modern societies in his study of liminoid phenomena in western society. He pointed out the similarities between the ‘leisure genres of art and entertainment in complex industrial societies and the rituals and myths of archaic, tribal and early agrarian cultures’ (1977:43).

Closely associated to liminality is communitas which describes a society during a liminal period that is ‘unstructured or rudimentarily structured [with] a relatively undifferentiated comitatus, community, or even communion of equal individuals who submit together to the general authority of the ritual elders’ (Turner, 1969:96).

The notion of communitas is enhanced by Turner’s concept of anti-structure. In the following passage Turner clarifies the ideas of liminal, communitas and anti-structure:

I have used the term ‘anti-structure,’… to describe both liminality and what I have called ‘communitas.’ I meant by it not a structural reversal… but the liberation of human capacities of cognition, affect, volition, creativity, etc., from the normative constraints incumbent upon occupying a sequence of social statuses (1982:44).

It is the potential of an anti-structured liminal person or liminal society (i.e., communitas) that makes Turner’s ideas so engaging. People or societies in a liminal phase are a ‘kind of institutional capsule or pocket which contains the germ of future social developments, of societal change’ (Turner, 1982:45).

Turner’s ideas on liminality and communitas have provided scholars with language to describe the state in which societal change takes place.”

Turner, V. (1967). The forest of symbols: Aspects of Ndembu ritual. Ithaca, NY: Cornell University Press.

Turner, V. (1969). The ritual process: structure and anti-structure. Chicago: Aldine Publishing Co.

Turner, V. (1977). Variations of the theme of liminality. In Secular ritual. Ed. S. Moore & B. Myerhoff. Assen: Van Gorcum, 36-52.

Turner, V. (1982). From ritual to theater: The human seriousness of play. New York: PAJ Publications.

Related material on Turner in Log24:

Aug. 27, 2006 and Aug. 30, 2006.  For further context, see archive of Aug. 19-31, 2006.

Related material on Cuernavaca:

Google search on Cuernavaca + Log24.

Friday, December 30, 2005

Friday December 30, 2005

Filed under: General — Tags: — m759 @ 1:00 am

Expression

Expression is an impossible word.
If you want to use it
I think you have to
explain it further…
— Ad Reinhardt,
Art as Art

“… an equation is a very
abstract expression
of knowledge
about something.”
The Hidden Side
of Visualization

Well, yes.
For example:

The image “http://www.log24.com/log/pix05B/051230-Shema.jpg” cannot be displayed, because it contains errors.
Abstract Expression
,
pixels on screen, 12/30/05

The Scope and Limits
of Quotation
(pdf), in
The Philosophy of
Donald Davidson,
ed. L. E. Hahn,
Open Court Publishers,
1999, pp. 691-714,
by Dr. Ernest Lepore,
Rutgers University:

“… an assumption I wish to reject, namely, that in quotation an abstract expression (shape) is denoted….

Since Davidson, however, only says ‘we may  take [an expression] to be an abstract shape’ [1979, p.85, my emphasis], his theory is compatible with expressions being something else. We need only find something that can be instantiated by radically differently shaped objects. Whatever it is must be such that written tokens, spoken tokens, signed tokens, Braille tokens, Semaphore tokens, finger language tokens, and any other way in which words can be produced, can be instantiated by it….

Such entities might  exist; if they do, they might ultimately play some role in the metaphysics of language.”

My emphasis.

Reference:

Davidson, D., 1979, “Quotation,” in Inquiries into Truth and Interpretation , Oxford, Oxford University Press, 1984, pp.79-92.

For more on zero and other entities,
see Is Nothing Sacred?

For more on Davidson,
see Shema .

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, April 7, 2005

Thursday April 7, 2005

Filed under: General — Tags: — m759 @ 9:00 am
Nine is a Vine

“Heaven is a state,
a sort of metaphysical state.”
— John O’Hara, Hope of Heaven, 1938

The image “http://www.log24.com/log/pix05/HopeOfHeaven1938.jpg” cannot be displayed, because it contains errors.

 “Mathematical realism holds that mathematical entities exist independently of the human mind.  Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato’s belief in a “heaven of ideas”, an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato’s view probably derives from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers. This idea may have even older origins that are unknown to us.” — Wikipedia

Amen.

Related material:

In memory of Jesus of Nazareth,
the “true vine,”
who, some historians believe,
died on this date:

The Crucifixion of John O’Hara.

In memory of the Anti-Vine:

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

See Dogma and
Heaven, Hell,
and Hollywood.
 
Related material:

The Usual Suspects

and

Thursday, December 26, 2002:

Holly for Miss Quinn 

Tonight’s site music is for Stephen Dedalus
and Miss Quinn, courtesy of Eithne Ní Bhraonáin. 

Miss Quinn

Holly

Eithne

Thursday, February 5, 2004

Thursday February 5, 2004

Filed under: General — m759 @ 12:00 pm

Affirmation of Place and Time:
East Coker and Grand Rapids

This morning’s meditation:

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

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

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

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

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

“Our emblem is
the cross in a triangle.”

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

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

by Karey Perkins

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

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

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

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

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

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

Thus the circle closed.

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

Monday, August 4, 2003

Monday August 4, 2003

Filed under: General — Tags: — m759 @ 1:00 pm

Venn's Trinity

Today is the birthday of logician John Venn.

From the St. Andrews History of Mathematics site:

"Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T." 

Last night's entry, "A Queer Religion," gave a Catholic view of the Trinity.  Here are some less interesting but more fruitful thoughts inspired by Venn's diagram of the Trinity (or, indeed, of any three entities):

"To really know a subject you've got to learn a bit of its history…."
John Baez, August 4, 2002

"We both know what memories can bring;
They bring diamonds and rust."
Joan Baez, April 1975

For the "diamonds" brought by memories of the 28 combinations described above, consider how the symmetric group S8 is related to the symmetries of the finite projective space PG(3,2).  (See Diamond Theory.) 

For the "rust," consider the following:

"Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt…."
— Matthew 6:19

The letters R, U, S, T in the Venn diagram above are perhaps relevant here, symbolizing, if you will, the earthly confusion of language, as opposed to the heavenly clarity of mathematics.

As for MOTH, see the article Hometown Zeroes (which brings us yet again to the Viper Room, scene of River Phoenix's death) and the very skillfully designed website MOTHEMATICS.

Saturday, July 12, 2003

Saturday July 12, 2003

Filed under: General — m759 @ 6:23 pm

Before and After

From Understanding the (Net) Wake:

24

A.

“Its importance in establishing the identities in the writer complexus….will be best appreciated by never forgetting that both before and after the Battle of the Boyne it was a habit not to sign letters always.”(114)

Joyce shows an understanding of the problems that an intertextual book like the Wake poses for the notion of authorship.

G. H. Hardy in A Mathematician’s Apology:

“We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases,’ indeed, is one of the duller forms of mathematical argument.  A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

A chess problem also has unexpectedness, and a certain economy; it is essential that the moves should be surprising, and that every piece on the board should play its part.  But the aesthetic effect is cumulative.  It is essential also (unless the problem is too simple to be really amusing) that the key-move should be followed by a good many variations, each requiring its own individual answer.  ‘If P-B5 then Kt-R6; if …. then …. ; if …. then ….’ — the effect would be spoilt if there were not a good many different replies.  All this is quite genuine mathematics, and has its merits; but it just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly*) which a real mathematician tends to despise.

* I believe that is now regarded as a merit in a problem that there should be many variations of the same type.”

(Cambridge at the University Press.  First edition, 1940.)

Brian Harley in Mate in Two Moves:

“It is quite true that variation play is, in ninety-nine cases out of a hundred, the soul of a problem, or (to put it more materially) the main course of the solver’s banquet, but the Key is the cocktail that begins the proceedings, and if it fails in piquancy the following dinner is not so satisfactory as it should be.”

(London, Bell & Sons.  First edition, 1931.)

Friday, March 21, 2003

Friday March 21, 2003

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

ART WARS:

Readings for Bach's Birthday

Larry J. Solomon:

Symmetry as a Compositional Determinant,
Chapter VIII: New Transformations

In Solomon's work, a sequence of notes is represented as a set of positions within a Latin square:

Transformations of the Latin square correspond to transformations of the musical notes.  For related material, see The Glass Bead Game, by Hermann Hesse, and Charles Cameron's sites on the Game.

Steven H. Cullinane:

Orthogonal Latin Squares as Skew Lines, and

Map Systems

Dorothy Sayers:

"The function of imaginative speech is not to prove, but to create–to discover new similarities, and to arrange them to form new entities, to build new self-consistent worlds out of the universe of undifferentiated mind-stuff." (Christian Letters to a Post-Christian World, Grand Rapids: Eerdmans, 1969, p. xiii)

— Quoted by Timothy A. Smith, "Intentionality and Meaningfulness in Bach's Cyclical Works"

Edward Sapir:

"…linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

 "The Grammarian and his Language,"
American Mercury 1:149-155, 1924

Tuesday, December 3, 2002

Tuesday December 3, 2002

Filed under: General — m759 @ 9:25 pm

From the Erlangen Program
to Category Theory

See the following, apparently all by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal:

See also the following by Marquis:

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