Log24

Thursday, March 1, 2018

The Movement of Analogy: Hume vs. Paz

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

Hume, from posts tagged "four-set" in this journal —

"The mind is a kind of theatre, where several perceptions
successively make their appearance; pass, repass, glide away,
and mingle in an infinite variety of postures and situations.
There is properly no simplicity in it at one time, nor identity
in different, whatever natural propension we may have
to imagine that simplicity and identity."

Paz, from a search for Paz + Identity in this journal —

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

Tuesday, October 17, 2017

The Movement of Analogy

Filed under: General,Geometry — Tags: — m759 @ 4:20 PM

The title is a phrase by Octavio Paz from today's post
"Status Symbols."

Other phrases from a link target in Sunday's post 
The Strength at the Centre

                               … a single world
In which he is and as and is are one.

See also Four Dots in this journal.

Monday, October 2, 2017

The Nut Analogy

Filed under: General — m759 @ 12:01 PM

For fans of the 'in a nutshell' quote from 'Hamlet'

Published as the final chapter, Chapter 13, in
Episodes in the History of Modern Algebra (1800-1950) ,
edited by Jeremy J. Gray and Karen Hunger Parshall,
American Mathematical Society, July 18, 2007,  pages 301-326.

See also this  journal on the above McLarty date —
May 24, 2003:  Mental Health Month, Day 24.

Wednesday, December 23, 2015

Analogy

Filed under: General — m759 @ 2:12 AM

"An analogy between mathematics and religion is apposite."

Harvard Magazine  review by Avner Ash of
     Mathematics without Apologies
    
(Princeton University Press, January 18, 2015)

See as well Analogies in this  journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: General,Geometry — Tags: , , — m759 @ 2:23 PM

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Monday, September 2, 2013

Analogy

Filed under: General — Tags: — m759 @ 10:30 PM

From The New York Times Sunday Book Review  of Sept. 1, 2013—

THE GAMAL
By Ciaran Collins
Illustrated. 469 pp. Bloomsbury. Paper, $17.

Reviewed by Katharine Weber

Ten years ago, when Mark Haddon’s “Curious Incident of the Dog in the Night-Time” turned up on the best-seller list and won a number of literary awards, the novel’s autistic narrator beguiled readers with his unconventional point of view. Today, even as controversy surrounds the revised classification of autism in the latest version of the American Psychiatric Association’s Diagnostic and Statistical Manual of Mental Disorders, the quirky yet remarkably perceptive points of view of autistic narrators have become increasingly familiar in every category of fiction, from young adult to science fiction to popular and literary fiction.

Like Haddon’s Christopher Boone, the narrator of Ciaran Collins’s remarkable first novel, “The Gamal,” has been encouraged by a mental health professional to write his story for therapeutic purposes. Charlie McCarthy, 25, is known in the West Cork village of Ballyronan as “the gamal,” short for “gamalog,” a term for a fool or simpleton rarely heard beyond the Gaeltacht regions of Ireland. He is in fact a savant, a sensitive oddball whose cheeky, strange, defiant and witty monologue is as disturbing as it is dazzling. …

The Gamal  features a considerable variety of music. See details at a music weblog.

This, together with the narrator's encouragement "by a mental health professional
to write his story for therapeutic purposes" might interest Baz Luhrmann.

See Luhrmann's recent film "The Great Gatsby," with its portrait of
F. Scott Fitzgerald's narrator, and thus Fitzgerald himself, as a sensitive looney.

The Carraway-Daisy-Gatsby trio has a parallel in The Gamal .  (Again, see
the music weblog's description.)  

The Times  reviewer's concluding remarks on truth, lies, and unreliable autistic
narrators may interest some mathematicians. From an Aug. 29 post

IMAGE- Barry Mazur: 'A good story is an end in itself.'

A different gamalog ,  a website in Mexico, is not entirely unrelated to
issues of lies and truth—

Wednesday, May 1, 2019

The Medium and the Message

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

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

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

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

 .

Monday, March 11, 2019

Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 PM

IMAGE- Concepts of Space

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract.

Wednesday, March 6, 2019

A Hand Calculator

Filed under: General — m759 @ 10:28 PM

Monday, September 17, 2018

Lying at the Axis

Filed under: G-Notes,General,Geometry — m759 @ 12:00 AM

Or:  Zero Dark Zero

" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "

— A footnote from page 229 of Sydney Padua's
    April 21, 2015, book on Lovelace and Babbage

Monday, August 20, 2018

A Wheel for Ellmann

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

The title was suggested by Ellmann's roulette-wheel analogy
in the previous post, "The Perception of Coincidence."

I Ching hexagrams as a Singer 63-cycle, plus zero

Friday, June 29, 2018

Analogies Between Analogies:

Filed under: General — m759 @ 3:33 AM

Literary Meditation for the Feast of  SS Peter and Paul

Background McLuhan on analogy.

See a publication offering facsimiles of the original 4×6 cards
of John Shade's "Pale Fire," as Nabokov described them.

Regarding these card proportions, note that 4/6 = 333/500, approximately —
the proportions of the text box in a post from yesterday.

"Continue a search for thirty-three and three" — Katherine Neville.

These rather pointless, but vaguely poetic, analogies were suggested by

  • Yesterday morning's "The Corrections," a post
    featuring spider ballooning and a dead poet, and
     
  • "Blue Dream," a post of Feb. 11, 2006.

Thursday, May 17, 2018

Leap

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

Quoted here on May 5, 2018

" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "

— A footnote from page 229 of Sydney Padua's
    April 21, 2015, book on Lovelace and Babbage

The page number  229 may also be interpreted, cabalistically,
as the date  2/29, Leap Day.

See Leap Day 2016 among the posts tagged Mind Spider.

Saturday, May 5, 2018

Galois Imaginary

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

" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "

— A footnote from page 229 of Sydney Padua's
    April 21, 2015, book on Lovelace and Babbage

A related passage —

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998) —

0

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space. 
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to .

i = an imaginary being

Here, on this complex space,
i  am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i  now rise to the text whenever anyone reads 
about Evariste Galois, preferring to remain 
just below the surface, 
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his 
vein and wrote an epic in red):
The text is a two-way mirror 
that allows me to look into
the life and times of the reader. 
Who knows, someday i  may rise
to a text that will compel me 
to push through to the other side.
Do you want proof that i  exist? Where am ?
Beneath every word, behind each letter, 
on the side of a period that will never see the light.

Wednesday, February 7, 2018

Conceptual Minimalism

Filed under: General,Geometry — m759 @ 10:08 AM

 

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

See also AS IS.

Wednesday, October 25, 2017

The Source (Not by Michener)

Filed under: General — m759 @ 11:18 AM
 

Wikipedia:  Taiji (philosophy)

Etymology

The word 太極 comes from I Ching : "易有太極,是生兩儀,兩儀生四象,四象生八卦,八卦定吉凶,吉凶生大業。"

Taiji  (太極) is a compound of tai   "great; grand; supreme; extreme; very; too" (a superlative variant of da   "big; large; great; very") and ji   "pole; roof ridge; highest/utmost point; extreme; earth's pole; reach the end; attain; exhaust". In analogy with the figurative meanings of English pole, Chinese ji  極 "ridgepole" can mean "geographical pole; direction" (e.g., siji  四極 "four corners of the earth; world's end"), "magnetic pole" (Beiji  北極 "North Pole" or yinji  陰極 "negative pole; cathode"), or "celestial pole" (baji  八極 "farthest points of the universe; remotest place"). Combining the two words, 太極 means "the source, the beginning of the world".

Common English translations of the cosmological Taiji  are the "Supreme Ultimate" (Le Blanc 1985, Zhang and Ryden 2002) or "Great Ultimate" (Chen 1989, Robinet 2008); but other versions are the "Supreme Pole" (Needham and Ronan 1978), "Great Absolute", or "Supreme Polarity" (Adler 1999).

Tuesday, October 17, 2017

Status Symbols

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

"Status: Defunct"  

As is now its owner, who reportedly
died at 80 on Sunday, October 15, 2017.

In memoriam —

Excerpts from Log24 posts on Sunday night 
and yesterday evening

http://www.log24.com/log/pix11/110203-Scholia.jpg.

" … listen: there's a hell
of a good universe next door; let's go"

— e. e. cummings

Some literary background —

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

Monday, June 26, 2017

Four Dots

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

Gian-Carlo Rota on Heidegger…

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

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

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

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

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

See also Four Dots in this journal. 

Some context:  McLuhan + Analogy.

Sunday, March 12, 2017

Raise High the Ridgepole, Architects*

Filed under: General — m759 @ 2:40 PM

A post suggested by remarks of J. D. Salinger in 
The New Yorker  of November 19, 1955 —

Wikipedia:  Taiji (philosophy)

Etymology

The word 太極 comes from I Ching : "易有太極,是生兩儀,兩儀生四象,四象生八卦,八卦定吉凶,吉凶生大業。"

Taiji  (太極) is a compound of tai   "great; grand; supreme; extreme; very; too" (a superlative variant of da   "big; large; great; very") and ji   "pole; roof ridge; highest/utmost point; extreme; earth's pole; reach the end; attain; exhaust". In analogy with the figurative meanings of English pole, Chinese ji  極 "ridgepole" can mean "geographical pole; direction" (e.g., siji  四極 "four corners of the earth; world's end"), "magnetic pole" (Beiji  北極 "North Pole" or yinji  陰極 "negative pole; cathode"), or "celestial pole" (baji  八極 "farthest points of the universe; remotest place"). Combining the two words, 太極 means "the source, the beginning of the world".

Common English translations of the cosmological Taiji  are the "Supreme Ultimate" (Le Blanc 1985, Zhang and Ryden 2002) or "Great Ultimate" (Chen 1989, Robinet 2008); but other versions are the "Supreme Pole" (Needham and Ronan 1978), "Great Absolute", or "Supreme Polarity" (Adler 1999).

See also Polarity in this journal.

* A phrase adapted, via Salinger,
from a poem by Sappho

Ἴψοι δὴ τὸ μέλαθρον,
     Υ᾽μήναον
ἀέρρετε τέκτονεσ ἄνδρεσ,
     Υ᾽μήναον
γάμβροσ ἔρχεται ἶσοσ Ά᾽ρευϊ,
     [Υ᾽μήναον]
ανδροσ μεγάλο πόλυ μείζων
     [Υ᾽μήναον]

Thursday, January 19, 2017

Inarticulate Image

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

The "inarticulate" image from last night's
"Raid on the Inarticulate" —

This is, in a sense, an island of nothing in a sea of being.

Contrast with an opposite image in Wittgenstein's "Diktat für Schlick":

From The Voices of Wittgenstein: The Vienna Circle
ed. by Gordon Baker, first published by Routledge 
in 2003. From Ch. 1, "Dictation for Schlick" —

p. 69 —
"Our method resembles psychoanalysis in a certain sense.
To use its way of putting things, we could say that a
simile at work in the unconscious is made harmless by
being articulated. And this comparison with analysis
p.71 —
can be developed even further. (And this analogy is
certainly no coincidence.)
     Anyone who speaks of the opposition of being and
the nothing, and of the nothing as something primary
in contrast to negation, has in mind, I think, a
picture of an island of being which is being washed
by an infinite ocean of the nothing. Whatever we throw
into this ocean will be dissolved in its water and
annihilated. But the ocean itself is endlessly restless
like the waves on the sea. It exists, it is, and we say
'It noths'. But how is it possible to demonstrate to
someone that this simile is actually the correct one?
This cannot be shown at all. But if we free him from his
confusion then we have accomplished what we wanted to
do for him."

"Ripples spread from castle rock …." — "Endgame," 1986

Thursday, October 6, 2016

Poetics

Filed under: General — Tags: — m759 @ 11:00 AM

Wednesday, August 10, 2016

Analogies Test

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

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

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

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

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

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

"You know nothing of my work."

Monday, August 8, 2016

A Point of Identity

Filed under: General — m759 @ 6:00 PM

For a  Monkey Grammarian  (Viennese Version)

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Shown below are a few variations on the figure by VCQ,
the Vienna Center for Quantum Science and Technology —
 

(Click image to enlarge.)

Thursday, July 14, 2016

Midnight Special

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

(Continued)

"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind.

… Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

A post  from this  journal later in 2010 —

The above post's date — May 20, 2010 — was
the date of death for mathematician Walter Rudin.

The above post from that date has a link to the
Heinlein story "And He Built a Crooked House."
A not-so-crooked house —

Friday, July 1, 2016

Transparent Core

Filed under: General — m759 @ 2:28 PM

"At the point of convergence the play of similarities and differences
cancels itself out in order that identity alone may shine forth.
The illusion of motionlessness, the play of mirrors of the one:
identity is completely empty; it is a crystallization and
in its transparent core the movement of analogy begins all over
once again." — The Monkey Grammarian  by Octavio Paz,
translated by Helen Lane 

A more specific "transparent core" —

See all references to this figure
in this journal.

For a more specific "monkey grammarian," 
see W. Tecumseh Fitch in this journal.

Thursday, June 2, 2016

Bullshit Studies

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

(Continued)

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

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

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

— McLuhan, ibid. , page 241

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

Epigraph by McLuhan —

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

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

Sunday, May 29, 2016

The Ideogram Principle …

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

According to McLuhan

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

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

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

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

A relation in four terms

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

See also Dueling Formulas and Symmetry.

Sunday School

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

Analogy

Thursday, April 14, 2016

Banach Revisited

Filed under: General — m759 @ 12:00 PM

A  1960  analogy by Max Black

"Those who see a model as a mere crutch
are like those who consider metaphor
a mere decoration or ornament."

This suggests a search for "Analogies between Analogies" —

“A mathematician is a person who can find analogies
between theorems; a better mathematician is one who
can see analogies between proofs and the best
mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one
who can see analogies between analogies.”

— Stefan Banach, according to MacTutor.

Monday, April 4, 2016

Cube for Berlin

Filed under: General,Geometry — m759 @ 11:00 AM

Foreword by Sir Michael Atiyah —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

 Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

"Like Rubik's Cube, The Eight  demands to be pondered."

As does a figure from 1984, Cullinane's Cube —

The Eightfold Cube

For natural group actions on the Cullinane cube, 
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 1984-2003."

(From Anatomy of a Cube, Sept. 18, 2011.)

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: — m759 @ 12:06 PM

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Friday, December 25, 2015

Dark Symbol

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

Related material:

The previous post (Bright Symbol) and
a post from Wednesday
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.

"An analogy between mathematics and religion is apposite."

— Harvard Magazine  review by Avner Ash of
     Mathematics without Apologies
     
(Princeton University Press, January 18, 2015)
 

Friday, December 18, 2015

Box of Nothing

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

Images related to the previous post

Detail of the 1697 Leibniz medal

Leibniz, letter of 1697:

“And so that I won’t come entirely empty-handed this time, I enclose a design of that which I had the pleasure of discussing with you recently. It is in the form of a memorial coin or medallion; and though the design is mediocre and can be improved in accordance with your judgment, the thing is such, that it would be worth showing in silver now and unto future generations, if it were struck at your Highness’s command. Because one of the main points of the Christian Faith, and among those points that have penetrated least into the minds of the worldly-wise and that are difficult to make with the heathen is the creation of all things out of nothing through God’s omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing. And it would be difficult to find a better illustration of this secret in nature or philosophy; hence I have set on the medallion design IMAGO CREATIONIS [in the image of creation]. It is no less remarkable that there appears therefrom, not only that God made everything from nothing, but also that everything that He made was good; as we can see here, with our own eyes, in this image of creation. Because instead of there appearing no particular order or pattern, as in the common representation of numbers, there appears here in contrast a wonderful order and harmony which cannot be improved upon….

Such harmonious order and beauty can be seen in the small table on the medallion up to 16 or 17; since for a larger table, say to 32, there is not enough room. One can further see that the disorder, which one imagines in the work of God, is but apparent; that if one looks at the matter with the proper perspective, there appears symmetry, which encourages one more and more to love and praise the wisdom, goodness, and beauty of the highest good, from which all goodness and beauty has flowed.”

See also some related posts in this journal.

Friday, May 8, 2015

Spielraum

Filed under: General,Geometry — Tags: — m759 @ 9:30 AM

Review:

Illustrating the Spiegel-Spiel des Gevierts

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by
Helen Lane 

 

Friday December 5, 2008

m759 @ 1:06 PM
 
Mirror-Play of
the Fourfold

For an excellent commentary
 on this concept of Heidegger,

View selected pages
from the book

Dionysus Reborn:

Play and the Aesthetic Dimension
in Modern Philosophical and
Scientific Discourse

(Mihai I. Spariosu,
Cornell U. Press, 1989)

Related material:
the logo for a
web page

Logo for 'Elements of Finite Geometry'

– and Theme and Variations.

Tuesday, May 5, 2015

Paz

Filed under: General — m759 @ 11:35 AM

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by
Helen Lane 

Tuesday, November 25, 2014

Euclidean-Galois Interplay

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

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

Saturday, May 11, 2013

Core

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

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

"A core component in the construction
is a 3-dimensional vector space  over F."

—  Page 29 of "A twist in the M24 moonshine story," 
      by Anne Taormina and Katrin Wendland.
      (Submitted to the arXiv on 13 Mar 2013.)

Monday, December 24, 2012

All Over Again

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

Octavio Paz —

"… the movement of analogy
begins all over once again."

See A Reappearing Number in this journal.

Illustrations:

Figure 1 —

Background: MOG in this journal.

Figure 2 —

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Background —

Image-- Google search on 'miracle octad'-- top 3 results

Sunday, December 23, 2012

Digital Recreation

Filed under: General — m759 @ 2:29 AM

Denzel Washington in Deja Vu  (2006), directed by Tony Scott—

Click to enlarge.

IMAGE- Denzel Washington in 'Deja Vu' (2006)

See also Tony Scott and four and a half days ago* —

  
Japanese character
       for "field"

Related material from five  days ago

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by 
Helen Lane (Kindle edition of 
2011-11-07, Kindle locations 
1207-1210).

* More precisely, what will be 4.5 days ago at 3:09 AM ET.

Friday, December 21, 2012

Analogies*

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

The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).

Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy between  the two above analogies
is supplied by the exceptional outer automorphism of S6.
See…

The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of  S6, and
A linear complex related to M24.

(Background: InscapesInscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)

* For some context, see Analogies and
  "Smallest Perfect Universe" in this journal.

Tuesday, December 18, 2012

Monkey Grammar

Filed under: General,Geometry — Tags: , — m759 @ 2:29 AM

For a modern Adam and Eve—

W. Tecumseh Fitch and Gesche Westphal Fitch,
editors of a new four-volume collection titled
Language Evolution  (Feb. 2, 2012, $1,360)—

Related material—

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by
Helen Lane (Kindle edition of
2011-11-07, Kindle locations
1207-1210).

The "play of mirrors" link above is my own.

Click on W. Tecumseh Fitch for links to some
examples of mirror-play in graphic design—
from, say, my own work in a version of 1977, not from
the Fitches' related work published online last June—

See also Log24 posts from the publication date
of the Fitches' Language Evolution

Groundhog Day, 2012.

Happy birthday to the late Alfred Bester.

Monday, May 21, 2012

Wittgenstein’s Kindergarten

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

A web search for the author Cameron McEwen  mentioned
in today's noon post was unsuccessful, but it did yield an
essay, quite possibly by a different  Cameron McEwen, on

The Digital Wittgenstein:

"The fundamental difference between analog
and digital systems may be understood as
underlying philosophical discourse since the Greeks."

The University of Bergen identifies the Wittgenstein 
McEwen as associated with InteLex  of Charlottesville.

The title of this post may serve to point out an analogy*
between the InteLex McEwen's analog-digital contrast
and the Euclidean-Galois contrast discussed previously
in this journal.

The latter contrast is exemplified in Pilate Goes to Kindergarten.

* An analogy, as it were, between  analogies.

Brightness at Noon

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

Occultation according to McLuhan

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

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

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

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

Friday, April 20, 2012

Complex Reflection

Filed under: General — m759 @ 11:32 AM

Yesterday's post in memory of Octavio Paz

… the free-standing, two-sided “Life-Death Figure,”
carved from stone in Mexico some time between
A.D. 900 and 1250, 
has multiple personalities.

Holland Cotter,  New York Times 

An earlier post yesterday, Fashion Notes, linked to a Sting video—

IMAGE- Sting meets his own reflection in a mirror in 'We'll Be Together' video

From "Loo Ree," by Zenna Henderson

"It's so hard to explain–"

"Oh, foof!" I cried defiantly, taking off my glasses and, smearing the tears across both lenses with a tattered Kleenex. "So I'm a dope, a moron! If I can explain protective coloration to my six-year-olds and the interdependence of man and animals, you can tell me something of what the score is!" I scrubbed the back of my hand across my blurry eyes. "If you have to, start out 'Once upon a time."' I sat down– hard.  

Loo Ree smiled and sat down, too. "Don't cry, teacher. Teachers aren't supposed to have tears."  

"I know it," I sniffed. "A little less than human-that's us."

"A little more than human, sometimes." Loo Ree corrected gently. "Well then, you must understand that I'll have to simplify. You will have to dress the bare bones of the explanation according to your capabilities.  

"Once upon a time there was a classroom. Oh, cosmic in size, but so like yours that you would smile in recognition if you could see it all. And somewhere in the classroom something was wrong. Not the whispering and murmuring– that's usual. Not the pinching and poking and tattling that goes on until you get so you don't even hear it." I nodded. How well I knew.  

"It wasn't even the sudden blow across the aisle or the unexpected wrestling match in the back of the room. That happens often, too. But something else was wrong. It was an undercurrent, a stealthy, sly sort of thing that has to be caught early or it disrupts the whole classroom and tarnishes the children with a darkness that will never quite rub off.  

"The teacher could feel it –as all good teachers can– and she spoke to the principal. He, being a good principal, immediately saw the urgency of the matter and also saw that it was beyond him, so he called in an Expert." "You?" I asked, feeling quite bright because I had followed the analogy so far.  

Loo Ree smiled. "Well, I'm part of the Expert."  

"If you have to, start out 'Once upon a time.'"

Yesterday's Paz post was at 6:48 PM EDT.

For the autistic, here is some related mathematics.

Yesterday's Fashion Notes post was at 1:06 PM  EDT.

A related chronological note from Rolling Stone  yesterday—

"Levon Helm, singer and drummer for the Band,
 died on April 19th in New York of throat cancer.
 He was 71. 

"He passed away peacefully at 1:30 this afternoon…."

Helm and The Band performing "The Weight"—

"I pulled into Nazareth, I was a-feelin' 'bout half past dead…"

Friday, July 29, 2011

Marshall, Meet Bagger

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

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

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

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

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

The field — See June 2010.

Sunday, June 13, 2010

Sunday School

Filed under: General — m759 @ 9:00 AM

What on earth is a 'concrete universal'?"
Said to be an annotation (undated)
by Robert M. Pirsig of A History of Philosophy,
by Frederick Copleston, Society of Jesus.

From Aaron Urbanczyk's 2005 review of Christ and Apollo  by William Lynch, S.J., a book first published in 1960—

"Lynch's use of analogy vis-a-vis literature provides, in a sense, a philosophical basis to the theoretical paradox popularized by W. K. Wimsatt (1907-1975), which contends that literature is a sort of 'concrete universal.'"

The following figure has often been
offered in this journal as a symbol of Apollo

Image-- 3x3 array of white squares

Arguments that it is, rather, a symbol of Christ
may be left to the Society of Jesus.

One possible approach—
Urbanczyk's review says that
"Christianity offers the critic
   a privileged ontological window…."

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

Delmore Schwartz

Sunday, April 18, 2010

A Generation Lost in Space

Filed under: General — m759 @ 2:29 AM

or, Deja Vu All Over Again

Top two obituaries in this morning's NY Times list–

David Simons, Who Flew High
on Eve of Space Age, Dies at 87

Dr. Simons, a physician turned Air Force officer, had sent animals aloft for several years before his record-breaking flight.

James Aubrey, who Portrayed the Hero
in ‘Lord of the Flies’, Is Dead at 62

Mr. Aubrey portrayed Ralph in the film version of the William Golding novel and had a busy career on stage and television in England.

Simons reportedly died on April 5,
Aubrey on April 6.

This journal on those dates–

April 5 —

Monday, April 5, 2010

Space Cowboys

Filed under: Uncategorized — m759 @ 12:00 PM Edit This

http://www.log24.com/log/pix10/100405-Eastwood.jpg

Google News, 11:32 AM ET today–

http://www.log24.com/log/pix10/100405-SpaceCowboysSm.jpg

Related material:

Yesterday's Easter message,
film notes from March 13,
and Dagger Definitions.

April 6 —

Tuesday, April 6, 2010

Clue

Filed under: Uncategorized — m759 @ 12:00 PM Edit This

Excerpt from 'Cosmic Trigger'
 by Robert Anton Wilson

See also Leary on Cuernavaca,
John O'Hara's fleeting reference
to Cuernavaca in Hope of Heaven,
and Cuernavaca in this journal.

Team Daedalus

Filed under: Uncategorized — m759 @ 9:00 AM Edit This

"Concept (scholastics' verbum mentis)– theological analogy of Son's procession as Verbum Patris, 111-12" –Index to Joyce and Aquinas, by William T. Noon, Society of Jesus, Yale University Press 1957, second printing 1963, page 162

"Back in 1958… [four] Air Force pilots were Team Daedalus, the best of the best." –Summary of the film "Space Cowboys"

"Man is nothing if not labyrinthine." –The Vicar in Trevanian's The Loo Sanction\

 

Commentary by T.S. Eliot

"At the moment which is not of action or inaction
You can receive this: 'on whatever sphere of being
The mind of a man may be intent
At the time of death'—that is the one action
(And the time of death is every moment)
Which shall fructify in the lives of others:
And do not think of the fruit of action.
Fare forward."

 

Tuesday, April 6, 2010

Team Daedalus

Filed under: General — m759 @ 9:00 AM

"Concept (scholastics' verbum mentis)– theological analogy of Son's procession as Verbum Patris, 111-12" –Index to Joyce and Aquinas, by William T. Noon, Society of Jesus, Yale University Press 1957, second printing 1963, page 162

"Back in 1958… [four] Air Force pilots were Team Daedalus, the best of the best." –Summary of the film "Space Cowboys"

"Man is nothing if not labyrinthine." –The Vicar in Trevanian's The Loo Sanction

Tuesday, March 16, 2010

Variations on a Theme

Filed under: General — Tags: — m759 @ 2:29 PM

Today's previous entry was "Gameplayers of the Academy."

More on this theme–

David Corfield in the March 2010
European Mathematical Society newsletter

    "Staying on the theme of games, the mathematician
Alexandre Borovik* once told me he thinks of mathematics
as a Massively-Multiplayer Online Role-Playing Game. If
so, it would show up very clearly the difference between
internal and external viewpoints. Inside the game people
are asking each other whether they were right about
something they encountered in it– 'When you entered
the dungeon did you see that dragon in the fireplace or
did I imagine it?' But someone observing them from the
outside wants to shout: 'You’re not dealing with anything
real. You’ve just got a silly virtual reality helmet on.' External
nominalists say the same thing, if more politely, to
mathematical practitioners. But in an important way the
analogy breaks down. Even if the players interact with
the game to change its functioning in unforeseen ways,
there were the original programmers who set the bounds
for what is possible by the choices they made. When they
release the next version of the game they will have made
changes to allow new things to happen. In the case of
mathematics, it’s the players themselves who make these
choices. There’s no further layer outside.
    What can we do then instead to pin down internal reality?"

*See previous references to Borovik in this journal.

Related material:

The Diamond Theory vs. the Story Theory of Truth,

Infantilizing the Audience, and

It's Still the Same Old Story…God of War III

Wednesday, January 6, 2010

Epiphany Revisited

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

January 06, 2007
ART WARS: Epiphany

Picture of Nothing
On Kirk Varnedoe’s
2003 Mellon Lectures,
Pictures of Nothing“–

“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….”

Related material:

The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.”

— Tom Wolfe, The Painted Word

Log24, Aug. 23, 2005:

“Concept (scholastics’ verbum mentis)–  theological analogy of Son’s procession  as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957,  second printing 1963, page 162

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

'In the beginning' according to Jim Holt

“Bang.”

“…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….”

For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.


Some Context:

Quaternions in Finite Geometry

Click to enlarge.

See also Nativity.

Friday, December 25, 2009

Brightness at Noon

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

New York Times online front page
Christmas morning:

“Arthur Koestler, Man of Darkness”–

NY Times front page, Christmas morning 2009

The photo is of Koestler in 1931 on a zeppelin expedition to the North Pole.

The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels….  According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ’see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”

– Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

From Opus Postumum by Immanuel Kant, Eckart Förster, Cambridge U. Press, 1995, p. 260:

“In January 1697, Leibniz accompanied his New Year Congratulations to Rudolf August with the design of a medal with the duke’s likeness on one side, and the ‘image of Creation’ in terms of the binary number system on the other. Concerning the inscription on this side, Leibniz writes: ‘I have thought for a while about the Motto dell’impresa and finally have found it good to write this line: omnibus ex nihilo ducendis SUFFICIT UNUM [To make all things from nothing, UNITY SUFFICES], because it clearly indicates what is meant by the symbol, and why it is imago creationis’ (G. F. Leibniz, Zwei Briefe über das binäre Zahlensystem und die chinesische Philosophie, ed. Renate Loosen and Franz Vonessen, Chr. Belser Verlag: Stuttgart 1968, p. 21).”

Leibniz, design for medallion showing binary numbers as an 'imago creationis'

Figure from Rudolf  Nolte’s
Gottfried Wilhelms Baron von Leibniz
Mathematischer Beweis der Erschaffung und
Ordnung der Welt in einem Medallion…
(Leipzig: J. C. Langenheim, 1734).

Leibniz, letter of 1697:

“And so that I won’t come entirely empty-handed this time, I enclose a design of that which I had the pleasure of discussing with you recently. It is in the form of a memorial coin or medallion; and though the design is mediocre and can be improved in accordance with your judgment, the thing is such, that it would be worth showing in silver now and unto future generations, if it were struck at your Highness’s command. Because one of the main points of the Christian Faith, and among those points that have penetrated least into the minds of the worldly-wise and that are difficult to make with the heathen is the creation of all things out of nothing through God’s omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing. And it would be difficult to find a better illustration of this secret in nature or philosophy; hence I have set on the medallion design IMAGO CREATIONIS [in the image of creation]. It is no less remarkable that there appears therefrom, not only that God made everything from nothing, but also that everything that He made was good; as we can see here, with our own eyes, in this image of creation. Because instead of there appearing no particular order or pattern, as in the common representation of numbers, there appears here in contrast a wonderful order and harmony which cannot be improved upon….

Such harmonious order and beauty can be seen in the small table on the medallion up to 16 or 17; since for a larger table, say to 32, there is not enough room. One can further see that the disorder, which one imagines in the work of God, is but apparent; that if one looks at the matter with the proper perspective, there appears symmetry, which encourages one more and more to love and praise the wisdom, goodness, and beauty of the highest good, from which all goodness and beauty has flowed.”

See also Parable.

Tuesday, July 14, 2009

Tuesday July 14, 2009

Filed under: General,Geometry — m759 @ 8:00 AM
For Galois on Bastille Day
 
Elements
of Finite Geometry


Some fans of the alchemy in
Katherine Neville’s novel
The Eight and in Dan Brown’s
   novel Angels & Demons may
  enjoy the following analogy

http://www.log24.com/log/pix09A/090714-Lattices.jpg

Note that the alchemical structure
at left, suited more to narrative
than to mathematics, nevertheless
 is mirrored within the pure
mathematics at right.

Related material
on Galois and geometry:

Geometries of the group PSL(2, 11)

by Francis Buekenhout, Philippe Cara, and Koen Vanmeerbeek. Geom. Dedicata, 83 (1-3): 169–206, 2000–

http://www.log24.com/log/pix09A/090714-Intro.jpg

Tuesday, March 10, 2009

Tuesday March 10, 2009

Filed under: General,Geometry — m759 @ 9:26 AM
Language Game

“Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns). The child-master, like his adult counterpart, is able to visualize in an instantaneous yet preternaturally confident way how the thing should look several moves hence. He sees the logical, the necessary harmonic and melodic argument as it arises out of an initial key relation or the preliminary fragments of a theme. He knows the order, the appropriate dimension, of the sum or geometric figure before he has performed the intervening steps. He announces mate in six because the victorious end position, the maximally efficient configuration of his pieces on the board, lies somehow ‘out there’ in graphic, inexplicably clear sight of his mind….”

“… in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra.”

— George Steiner, “A Death of Kings,” in The New Yorker, issue dated Sept. 7, 1968

Related material:

“Classrooms are filled with discussions not of the Bible and Jesus but of 10 ‘core values’– perseverance and curiosity, for instance– that are woven into the curriculum.”

— “Secular Education, Catholic Values,” by Javier C. Hernandez, The New York Times, Sunday, March 8, 2009

“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”


— Raymond Chandler, The Big Sleep

The Chandler quotation appears in “Language Game,” an entry in this journal on April 7, 2008.

Some say the “Language Game” date, April 7, is the true date (fixed, permanent) of the Crucifixion– by analogy, Eliot’s “still point” and Jung’s “centre.” (See yesterday, noon.)

Monday, March 9, 2009

Monday March 9, 2009

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

Humorism

'The Manchurian Candidate' campaign button

"Always with a
little humor."
Dr. Yen Lo  

Diamond diagram of the four humors, the four qualities, the four elements, the four seasons, and four colors

From Temperament: A Brief Survey

For other interpretations
of the above shape, see
The Illuminati Diamond.

from Jung's Aion:

"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung, Collected Works, Vol. 9, Part II, paragraph 352

As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison
 
Click on image
for a related puzzle.
For a solution, see
 The Diamond Theorem.

A related note on
"Angels & Demons"
director Ron Howard:

Director Ron Howard with illustration of the fictional discipline 'symbology'
 
Click image for details.

Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
 
Hollywood Nihilism
Meets
Pantheistic Solipsism

Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
 to hear about our religion
… that we made up."

Tina Fey and Steve Martin at the 2009 Oscars

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:

… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer

 A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.


Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."

As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.

Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.

Heinlein:

"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
    I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."

Stevens:

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:

B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":

"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'….  Its subject is its speaker's sense of nothingness and his need to be cured of it."

This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism.

More positively…

Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space
(or the corresponding
5-dimensional projective space)

The 4x4x4 cube

over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."

Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.

Cara:

Philippe Cara on the Klein correspondence
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:30 AM
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

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

Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Another possibility —
 

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
 

Tuesday, February 26, 2008

Tuesday February 26, 2008

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM
The Just Word

The title of the previous entry, "Where Entertainment is God," comes (via Log24, Nov. 26, 2004) from Frank Rich.

The previous entry dealt, in part, with a dead Jesuit whose obituary appears in today's Los Angeles Times.  The online obituaries page places the Jesuit, without a photo, beneath a picture of a dead sitcom writer and to the left of a picture of a dead guru.

From the obituary proper:

Walter J. Burghardt, alleged preacher of 'the just word'

The obituary does not say
exactly what "the just word" is.
 

"Walter John Burghardt was born July 10, 1914, in New York, the son of immigrants from what is now Poland. He entered a Jesuit seminary in Poughkeepsie, N.Y., at 16, and in 1937 received a master's degree from Woodstock College in Maryland. He was ordained in 1941." He died, by the way, on Saturday, Feb. 16, 2008.

The reference to Woodstock College brings to mind a fellow Jesuit, Joseph T. Clark, who wrote a book on logic published by that college.

From a review of the book:

"In order to show that Aristotelian logicians were at least vaguely aware of a kind of analogy or possible isomorphism between logical relations and mathematical relations, Father Clark seizes at one place (p. 8) upon the fact that Aristotle uses the word, 'figure' (schema), in describing the syllogism and concludes from this that 'it is obvious that the schema of the syllogism is to serve the logician precisely as the figure serves the geometer.' On the face of it, this strikes one as a bit far fetched…."

Henry Veatch in Speculum, Vol. 29, No. 2, Part 1 (Apr., 1954), pp. 266-268 (review of Conventional Logic and Modern Logic: A Prelude to Transition (1952), by Joseph T. Clark, Society of Jesus)
 

Perhaps the just word is,
as above, "schema."

Related material:

The Geometry of Logic

Thursday, October 25, 2007

Thursday October 25, 2007

Filed under: General,Geometry — Tags: — m759 @ 9:19 AM

Something Anonymous

From this date–
Picasso's birthday–
five years ago:
 
"A work of art has an author
and yet,
when it is perfect,
it has something
which is
essentially anonymous about it."

Simone Weil, Gravity and Grace   

 
Michelangelo's birthday, 2003

4x4 square grid

Yesterday:

The color-analogy figures of Descartes

Nineteenth-century quilt design:

Tents of Armageddon quilt design

Related material:

Battlefield Geometry
 

Wednesday, October 24, 2007

Wednesday October 24, 2007

Filed under: General,Geometry — Tags: , — m759 @ 11:11 PM
 
Descartes's Twelfth Step

An earlier entry today ("Hollywood Midrash continued") on a father and son suggests we might look for an appropriate holy ghost. In that context…

Descartes

A search for further background on Emmanuel Levinas, a favorite philosopher of the late R. B. Kitaj (previous two entries), led (somewhat indirectly) to the following figures of Descartes:

The color-analogy figures of Descartes
This trinity of figures is taken from Descartes' Rule Twelve in Rules for the Direction of the Mind. It seems to be meant to suggest an analogy between superposition of colors and superposition of shapes.Note that the first figure is made up of vertical lines, the second of vertical and horizontal lines, and the third of vertical, horizontal, and diagonal lines. Leon R. Kass recently suggested that the Descartes figures might be replaced by a more modern concept– colors as wavelengths. (Commentary, April 2007). This in turn suggests an analogy to Fourier series decomposition of a waveform in harmonic analysis. See the Kass essay for a discussion of the Descartes figures in the context of (pdf) Science, Religion, and the Human Future (not to be confused with Life, the Universe, and Everything).

Compare and contrast:

The harmonic-analysis analogy suggests a review of an earlier entry's
link today to 4/30–  Structure and Logic— as well as
re-examination of Symmetry and a Trinity


(Dec. 4, 2002).

See also —

A Four-Color Theorem,
The Diamond Theorem, and
The Most Violent Poem,

Emma Thompson in 'Wit'

from Mike Nichols's birthday, 2003.

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

Saturday, June 30, 2007

Saturday June 30, 2007

Filed under: General — m759 @ 1:00 PM
An Evening Star

for Rabbi Abraham Klausner,
a “father figure” according to

The New York Times.
The Times says Klausner
died at 92 on
Thursday, June 28, 2007:

(Click to enlarge.)

Rabbi Abraham Klausner

Klausner was a rabbi
in Yonkers until his
retirement in 1989.
The evening number in
the New York Lottery
on the reported date of
Klausner’s death
was 514.

As in the previous entry,
this number may be
interpreted as the date 5/14.

A Log24 entry with that date
:

 

Sunday, May 14, 2006

Today’s birthday: George Lucas,
creator of the mother of all battle epics.

STAR WARS continued:

March 29 eclipse
Star of Venus
Star of Venus
(See March 26-29)

In the details:


The image “http://www.log24.com/log/pix07/070630-Detail.jpg” cannot be displayed, because it contains errors.

Clicking on “Joshua” will take you
to a site on a film opening
July 6.  That site describes

the title character as follows:
 
“Joshua is no ordinary boy….

He’s exceptionally intelligent and frighteningly precocious.

He has an angelic politeness and an easy cool that belie his young age….

Is it all a series of eerie coincidences or are they in the midst of an unimaginably evil mind? And could it be Joshua who, like his Biblical namesake, is bringing the house tumbling down around his family?”

The “Biblical namesake” is the
Joshua of the Old Testament–
source of the deeply flawed
“tumbling down” analogy.

In the New Testament,

there is of course also
a rather famous Joshua.

The image “http://www.log24.com/log/pix07/070630-FoxLogo.jpg” cannot be displayed, because it contains errors.

“And the serpent’s eyes shine  
   as he wraps around the vine….”

The Garden of Allah

 

Thursday, June 14, 2007

Thursday June 14, 2007

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
A Flag for Sunset

"Kurt Waldheim, the former United Nations Secretary General and President of Austria whose hidden ties to Nazi organizations and war crimes was [sic] exposed late in his career, died today at his home in Vienna. He was 88." —The New York Times this afternoon
 

Related material:

From a story by
Leonard Michaels
linked to on
Aaron Sorkin's
birthday, June 9:

"Induction and analogy, in which he was highly gifted, were critical to mathematical intelligence.

It has been said that the unexamined life isn't worth living. Nachman wasn't against examining his life, but then what was a life? ….

… As for 'a life,' it was what you read about in newspaper obituaries. He didn't need one. He would return to California and think only about mathematics."

Mathematics:

1.  A quotation from George Polya,
     the author of
     Induction and Analogy
     in Mathematics

2.  A quotation from an anonymous
     Internet user signed
     "George Polya"–
     "Steven Cullinane is a Liar."

3.  L'Affaire Dharwadker continues
     (May 31, 2007)

4.  Geometry for Jews

The image “http://log24.com/theory/images/070614-Whirl.gif” cannot be displayed, because it contains errors.

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

 

Thursday, June 7, 2007

Thursday June 7, 2007

Filed under: General — m759 @ 4:15 PM
Framing
truth

On “framing” and “spin”
in journalism:

“… Packaging is unavoidable.
Facts rarely, if ever, 
  speak for themselves.”

Matthew C. Nisbet,  
Assistant Professor
  of “Communication,”
June 6, 2007

If they could, they might
say “We was framed!”

Facts cannot, of course,
speak for themselves
to those who do not
understand their language.

Example:

A picture that appeared in
Log24 on June 7, 2005:

Natural Transformation

Click for details.

Attempt to
frame the picture:

Analogies

“A functor is an analogy.”
— Anonymous

  The best mathematicians “see
analogies between analogies.”
Banach, according to Ulam 

For further details,
click on the link
“Analogies” above.

See also the analogies in
the previous entry.

Monday, January 29, 2007

Monday January 29, 2007

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

By Indirections
(Hamlet, II, i)

“Michael Taylor (1971)…. contends that the central conflict in Hamlet is between ‘man as victim of fate and as controller of his own destiny.'”– The Gale Group, Shakespearean Criticism, Vol. 71, at eNotes

Doonesbury today:

The image “http://www.log24.com/log/pix07/070129-Robot4A.gif” cannot be displayed, because it contains errors.

“Personality is a synthesis of possibility and necessity.”– Soren Kierkegaard

On Fate (Necessity),
Freedom (Possibility),
and Machine Personality–


Part I: Google as Skynet

George Dyson–
The Godel-to-Google Net [March 8, 2005]
A Cathedral for Turing [October 24, 2005]

Dyson: “The correspondence between Google and biology is not an analogy, it’s a fact of life.”

Part II: The Galois Connection

David Ellerman–
“A Theory of Adjoint Functors– with some Thoughts about their Philosophical Significance” (pdf) [November 15, 2005]

Ellerman: “Such a mechanism seems key to understanding how an organism can perceive and learn from its environment without being under the direct stimulus control of the environment– thus resolving the ancient conundrum of receiving an external determination while exercising self-determination.”

For a less technical version, see Ellerman’s “Adjoints and Emergence: Applications of a New Theory of Adjoint Functors” (pdf).

Ellerman was apparently a friend of, and a co-author with, Gian-Carlo Rota.  His “theory of adjoint functors” is related to the standard mathematical concepts known as profunctors, distributors, and bimodules. The applications of his theory, however, seem to be less to mathematics itself than to a kind of philosophical poetry that seems rather closely related to the above metaphors of George Dyson. For a less poetic approach to related purely mathematical concepts, see, for instance, the survey Practical Foundations of Mathematics by Paul Taylor (Cambridge University Press, 1999).  For less poetically appealing, but perhaps more perspicuous, extramathematical applications of category theory, see the work of, for instance, Joseph Goguen: Algebraic Semiotics and Information Integration, Databases, and Ontologies.

Saturday, January 6, 2007

Saturday January 6, 2007

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


Picture of Nothing

On Kirk Varnedoe’s
2003 Mellon Lectures,
Pictures of Nothing“–

“Varnedoe’s lectures were ultimately
about faith, about his faith in
the power of abstraction,
and abstraction as a kind of
anti-religious faith in itself….”

The Washington Post

Related material:

The more industrious scholars
will derive considerable pleasure
from describing how the art-history
professors and journalists of the period
1945-75, along with so many students,
intellectuals, and art tourists of every
sort, actually struggled to see the
paintings directly, in the old
pre-World War II way,
like Plato’s cave dwellers
watching the shadows, without
knowing what had projected them,
which was the Word.”

— Tom Wolfe, The Painted Word

Log24, Aug. 23, 2005:

“Concept (scholastics’ verbum mentis)–
theological analogy of Son’s procession
as Verbum Patris, 111-12″

— Index to Joyce and Aquinas,
by William T. Noon, S.J.,
Yale University Press 1957,
second printing 1963, page 162

“So did God cause the big bang?
Overcome by metaphysical lassitude,
I finally reach over to my bookshelf
for The Devil’s Bible.
Turning to Genesis I read:
‘In the beginning
there was nothing.
And God said,
‘Let there be light!’
And there was still nothing,
but now you could see it.'”

— Jim Holt, Big-Bang Theology,
Slate‘s “High Concept” department

The image “http://www.log24.com/log/pix07/070106-Bang.jpg” cannot be displayed, because it contains errors.


“Bang.”

“…Mondrian and Malevich
are not discussing canvas
or pigment or graphite or
any other form of matter.
They are talking about
Being or Mind or Spirit.
From their point of view,
the grid is a staircase
to the Universal….”

Rosalind Krauss, “Grids”

For properties of the
“nothing” represented
by the 3×3 grid, see
The Field of Reason.

For religious material related
to the above and to Epiphany,
a holy day observed by some,
see Plato, Pegasus, and the
Evening Star
and Shining Forth.

Sunday, July 2, 2006

Sunday July 2, 2006

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

The Rock and the Serpent

In a search for a title to express
the contrast between truth and lies,
an analogy between the phrases

Crystal and Dragon” and
Mathematics and Narrative

suggests a similar phrase,

“The Rock and the Serpent.”

A web search for related titles leads to a book by Alice Thomas Ellis:

Serpent on the Rock: A Personal View of Christianity. (See a review.)

(This in turn leads to an article on Ellis’s husband, the late Colin Haycraft, publisher.)

For an earlier discussion of Ellis in this weblog, see Three Eleanors (March 12, 2005).

That entry brings us back to the theme of truth and lies with its link to an article from the Catholic publication Commonweal:

Getting to Truth by Lying.

Christians who wish to lie more effectively may consult a book by the author of the Commonweal article:

The image “http://www.log24.com/log/pix05/050312-Form.jpg” cannot be displayed, because it contains errors.

For a more sympathetic view of
suffering stemming from
Christian narrative,
see

The image “http://www.log24.com/log/pix06A/LecturesOnDonQuixote.gif” cannot be displayed, because it contains errors.

(Click on cover
for details. See also Log24
entries on Guy Davenport,
who wrote the foreword.)

Wednesday, May 10, 2006

Wednesday May 10, 2006

Filed under: General,Geometry — m759 @ 4:29 PM
My Space

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

“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”

— Robert A. Heinlein,
The Number of the Beast

The image “http://www.log24.com/theory/images/Screenshot-64cube.gif” cannot be displayed, because it contains errors.

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

And so…

Compare and contrast:

Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.

Related material:

A more extensive excerpt from
The Number of the Beast, and

Story Theory and
the Number of the Beast.

Saturday, December 24, 2005

Saturday December 24, 2005

Filed under: General — m759 @ 12:00 PM
High Concept

“Concept (scholastics’ verbum mentis)–
theological analogy of Son’s procession
as Verbum Patris, 111-12″
— index to Joyce and Aquinas,    
by William T. Noon,
Society of Jesus,
Yale University Press 1957,
second printing 1963, page 162

Then there is
the Daughter’s procession:

The image “http://www.log24.com/log/pix05B/051224-String.gif” cannot be displayed, because it contains errors.

For the String Theory
Appreciation Club, see
  Raoul Bott, 1923-2005.

For another
imaginary club, see
The Club Dumas (below).

For a non-imaginary club,
see the organization
that included Noon (above).

Tuesday, November 1, 2005

Tuesday November 1, 2005

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM
The image “http://www.log24.com/log/pix05B/051101-Seal.gif” cannot be displayed, because it contains errors.

 

The above seal is from an ad (pdf) for an Oct. 21 lecture, "The Nature of Space," by Sir Michael Atiyah, sponsored by the American Mathematical Society.

The picture in the seal is of Plato's Academy.

"The great philosopher Plato excluded from his Academy anyone who had not studied geometry.  He would have been delighted to admit Sir Michael Atiyah, who was for a time Savilian Professor of Geometry at Oxford…"

 

Would he?

Sir Michael Atiyah's
Anti-Platonism

"Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance….

I tend to think that science and mathematics are ways the human mind looks and experiences– you cannot divorce the human mind from it. Mathematics is part of the human mind. The question whether there is a reality independent of the human mind, has no meaning– at least, we cannot answer it."

— Sir Michael Atiyah, interview in Oslo, May 2004

"For Plato, the Forms represent truth, or reality…. these Forms are independent of the mind: they are eternal, unchanging and perfect."

—  Roy Jackson (pdf)

Atiyah's denial of a reality independent of the human mind may have something to do with religion:

"Socrates and Plato were considered 'Christians before Christ'; they paved the way for the coming of Christianity by providing it with philosophical and theoretical foundations that would be acceptable to the western mind.
    In the analogy of the cave, the sun represents the Form of the Good. In the same way that the sun is the source of all things and gives light to them, the Form of the Good is over and above the other Forms, giving them light and allowing us to perceive them. Therefore, when you have awareness of the Form of the Good you have achieved true enlightenment. In Christianity, the Form of the Good becomes God: the source of all things."

— Roy Jackson, The God of Philosophy (pdf)

See also the previous entry.
 

Saturday, September 24, 2005

Saturday September 24, 2005

Filed under: General — m759 @ 11:02 PM
Parable

From this week’s New Yorker
and from Eight is a Gate:

The image “http://www.log24.com/log/pix05B/050924-NYer.jpg” cannot be displayed, because it contains errors.

May 20, 2004, 7 AM:

Parable: “A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’)….”

A Synoptic
   Gospels Primer

A thought dated (mistakenly)
May 20, 2004,
11:11 PM:

Life changes fast.

— Joan Didion,
  After Life

Related material:

Key,
Number 61,
Chorus from “The Rock”.

Tuesday, August 23, 2005

Tuesday August 23, 2005

Filed under: General — m759 @ 12:00 PM

High Concept*

“Concept (scholastics’ verbum mentis)–
 theological analogy of Son’s procession
 as Verbum Patris, 111-12″
 — index to Joyce and Aquinas,
 by William T. Noon, S.J.,
Yale University Press 1957,
 second printing 1963, page 162

“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”

— Jim Holt, Big-Bang Theology, Slate‘s “High Concept” department

Related material:

Nothing Ventured,
The God-Shaped Hole, and
Is Nothing Sacred?

 * See also John O’Callaghan, Thomistic Realism and the Linguistic Turn: Toward a More Perfect Form of Existence, (University of Notre Dame Press, 2003) and Joshua P. Hochschild, “Does Mental Language Imply Mental Representationalism? The Case of Aquinas’s Verbum Mentis,” Proceedings of the Society for Medieval Logic and Metaphysics, Volume 4, 2004 (pdf), pp. 12-17.

Thursday, June 23, 2005

Thursday June 23, 2005

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

Mathematics and Metaphor

The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles.  The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible.  The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history  and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
 
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.

Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated — 

“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182

This is false.  See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.

It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —

From Peter Woit’s weblog, Feb. 10, 2005:

“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'” 

In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that

“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two.  And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….

On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”

Here is another statement of the Rosetta-stone metaphor, from Weil’s translator, Martin H.  Krieger, in the A.M.S. Notices of November 2004,  “Some of What Mathematicians Do” (pdf):

“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’  Given an ability to read one column, can you find its translation in the other columns?  In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry.  In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”

For greater clarity, see  Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraico-geometric,” over finite fields), and arithmetic (i.e., number-theoretic).
 
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:

Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world.  Both these branches of mathematics had been studied intensively but separately…. Than along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds.  Mathematicians love to build bridges.”

Simon Singh: “The value of mathematical bridges is enormous.  They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s  creations…. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time.  Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur.  ‘But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'”

If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.

Saturday, June 11, 2005

Saturday June 11, 2005

Filed under: General — m759 @ 7:10 PM
Picture This

In memory of film producer Fernando Ghia:

“Among Ghia’s solo credits as a producer is

The image “http://www.log24.com/log/pix05A/050611-Lamb3.jpg” cannot be displayed, because it contains errors.

Lady Caroline Lamb,’ a 1972 period drama
written and directed by Robert Bolt.”

Today’s LA Times

Ghia died on June 1, 2005
(the date of the Dutch “No” vote).
In the spirit of Pale Fire, here is an excerpt
from a Log24 entry of that date:

The Road to Brussels


“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”

 — Henry Kissinger, quoted in
     Drama of the Diagonal, Part Deux

Les livres d’histoire et la vie
racontent la même comédie….

Alain Boublil




“Along the road from Ohain to Braine-l’Alleud that hemmed in the plain of Mont-St-Jean and cut at right angles the road to Brussels, which the Emperor wished to take, he [Wellington] had placed 67,000 men and 184 cannons.” Fr. Libert, Waterloo

In researching this entry, I thought of
Wellington’s statement
in “Lady Caroline Lamb” —
These are the Scots Greys.”

The image “http://www.log24.com/log/pix05A/050601-Forever.jpg” cannot be displayed, because it contains errors.

and found the above picture.

Related material:

Women’s History Month.

Saturday June 11, 2005

Filed under: General — m759 @ 3:11 AM

The Last Word

Beethoven Week on the BBC ended at midnight June 10.

“With Beethoven, music did not grow up, it regressed to adolescence. He was a hooligan who could reduce Schiller’s Ode to Joy to madness, bloodlust, and megalomania.”

Arts and Letters Daily, lead-in to an opinion piece in The Guardian of Tuesday, June 7, 2005:

Beethoven Was a Narcissistic Hooligan

“If Beethoven had dedicated his obvious talents to serving the noble Pythagorean view of music, he might well have gone on to compose music even greater than that of Mozart. You can hear this potential in his early string quartets, where the movements often have neat conclusions and there is a playfulness reminiscent of Mozart or Haydn. If only Beethoven had nourished these tender shoots instead of the darker elements that one can also hear. For the darkness is already evident in the early quartets too, in their sombre harmonies and sudden key changes. As it was, however, his darker side won out; compare, for example, the late string quartets. Here the youthful humour has completely vanished; the occasional signs of optimism quickly die out moments after they appear and the movements sometimes end in uncomfortably inconclusive cadences….

In A Clockwork Orange it is the fourth movement of Beethoven’s Ninth Symphony that echoes in the mind of Alex whenever he indulges in one of his orgies of violence. Alex’s reaction may be rather extreme, but he is responding to something that is already there in this dark and frenzied setting of Schiller’s Ode to Joy; the joy it invites one to feel is the joy of madness, bloodlust and megalomania. It is glorious music, and seductive, but the passions it stirs up are dark and menacing.”

The image “http://www.log24.com/log/pix05A/050611-Clock.jpg” cannot be displayed, because it contains errors.

Dylan Evans, former Lacanian psychotherapist (pdf) and now head of the undergraduate robotics program at the University of the West of England.

Speak for yourself, Dylan.

“Evil did not have the last word.”

—  Richard John Neuhaus, April 4, 2005

Evil may have had the last word in Tuesday’s Guardian, but now that Beethoven Week has ended, it seems time for another word.

For another view of Beethoven, in particular the late quartets, see the Log24 Beethoven’s Birthday entry of December 16, 2002:

Beethoven’s Birthday

“Ludwig van Beethoven’s String Quartet in A Minor, Opus 132, is one of the transcendent masterworks of the Western classical tradition. It is built around its luminous third movement, titled ‘Holy song of thanksgiving by one recovering from an illness.’

In this third movement, the aging Beethoven speaks, clearly and distinctly, in a voice seemingly meant both for all the world and for each individual who listens to it. The music, written in the ancient Lydian mode, is slow and grave and somehow both a struggle and a celebration at the same time.

This is music written by a supreme master at the height of his art, saying that through all illness, tribulation and sorrow there is a strength, there is a light, there is a hope.”

—  Andrew Lindemann Malone

“Eliot’s final poetic achievement—and, for many, his greatest—is the set of four poems published together in 1943 as Four Quartets…. Structurally—though the analogy is a loose one—Eliot modeled the Quartets on the late string quartets of Beethoven, especially… the A Minor Quartet; as early as 1931 he had written the poet Stephen Spender, ‘I have the A Minor Quartet on the gramophone, and I find it quite inexhaustible to study. There is a sort of heavenly or at least more than human gaiety about some of his later things which one imagines might come to oneself as the fruit of reconciliation and relief after immense suffering; I should like to get something of that into verse before I die.'”

— Anonymous author at a
Longman Publishers website

“Each of the late quartets has a unique structure, and the structure of the Quartet in A Minor is one of the most striking of all. Its five movements form an arch. At the center is a stunning slow movement that lasts nearly half the length of the entire quartet…

The third movement (Molto adagio) has a remarkable heading: in the score Beethoven titles it ‘Hymn of Thanksgiving to the Godhead from an Invalid,’ a clear reflection of the illness he had just come through. This is a variation movement, and Beethoven lays out the slow opening section, full of heartfelt music. But suddenly the music switches to D major and leaps ahead brightly; Beethoven marks this section ‘Feeling New Strength.’ These two sections alternate through this movement (the form is A-B-A-B-A), and the opening section is so varied on each reappearance that it seems to take on an entirely different character each time: each section is distinct, and each is moving in its own way (Beethoven marks the third ‘With the greatest feeling’). This movement has seemed to many listeners the greatest music Beethoven ever wrote. and perhaps the problem of all who try to write about this music is precisely that it cannot be described in words and should be experienced simply as music.”

—  Eric Bromberger,
Borromeo Quartet program notes 

In accordance with these passages, here is a web page with excellent transcriptions for piano by Steven Edwards of Beethoven’s late quartets:

The 16 String Quartets.

Our site music for today, Beethoven’s String Quartet No. 15 in A Minor, Opus 132, Movement 3 (1825), is taken from this web page.

See also the previous entry.
 

Saturday, June 4, 2005

Saturday June 4, 2005

Filed under: General,Geometry — m759 @ 7:00 PM
  Drama of the Diagonal
  
   The 4×4 Square:
  French Perspectives

Earendil_Silmarils:
The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.
  
   Les Anamorphoses:
 
   The image “http://www.log24.com/log/pix05A/050604-DesertSquare.jpg” cannot be displayed, because it contains errors.
 
  “Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d’horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4).”
   _______________________________
  
  Serge Mehl,
   Perspective &
  Géométrie Projective:
  
   “… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s’opposait à la géométrie
   euclidienne: élémentaire
  
  La géométrie projective, certes supérieure
   car assez ardue, permet d’établir
   de façon élégante des résultats de
   la géométrie élémentaire.”
  
  Similarly…
  
  Finite projective geometry
  (in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
  quilt-pattern symmetry
  and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.
  
  Other Related Material…
  
   from algebra rather than
   geometry, and from a German
   rather than from the French:  

This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946

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

Evariste Galois

 Weyl also says that the profound branch
of mathematics known as Galois theory

   “… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
    finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory.”
  — Weyl, Symmetry,
   Princeton U. Press, 1952
  
   Metaphor and Algebra…  

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

   — attributed, in varying forms, to
   Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see  

  “Symmetry invariance
  in a diamond ring,”

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

  
More on Max Black…

“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”

— Paul Thompson, University College, Oxford,
    The Nature and Role of Intuition
     in Mathematical Epistemology

  A New Slant…  

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known.  The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.”  The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

Wednesday, June 1, 2005

Wednesday June 1, 2005

Filed under: General — m759 @ 7:20 PM
The Road to Brussels


“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”

 — Henry Kissinger, quoted in
     Drama of the Diagonal, Part Deux

Les livres d’histoire et la vie
racontent la même comédie….

Alain Boublil


Wellington at Waterloo


The image “http://www.log24.com/log/pix05/050601-Map.jpg” cannot be displayed, because it contains errors.

livgenmi.com/gardiner85.htm 


“Along the road from Ohain to Braine-l’Alleud that hemmed in the plain of Mont-St-Jean and cut at right angles the road to Brussels, which the Emperor wished to take, he [Wellington] had placed 67,000 men and 184 cannons.” Fr. Libert, Waterloo

The Emperor’s Welcome

From Expatriate Online:
Your Bookmark to Belgium

The image “http://www.log24.com/log/pix05/050601-Waterloo.jpg” cannot be displayed, because it contains errors.

Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 PM
Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares
(ps),
a discussion of a 4×4 array
of square linear designs.
  Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle:  the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16.  The first group is the most obvious.  The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
  Being and Nothingness,
  Philosophical Library, 1956
  [reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
 

  The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1. on Page 51:
“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being.  Man is ‘a being of distances.'”
2. on Page 154:
“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities.  Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3. on Page 155:
“… end of the desires.  But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the  glass-in-the-midst-of-the-world.”
4. on Page 158:
“…  it is in time that my possibilities appear on the horizon of the world which they make mine.  If, then, human reality is itself apprehended as temporal….”
5. on Page 180:
“… else time is an illusion and chronology disguises a strictly logical order of  deducibility.  If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6. on Page 186:
“…  It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7. on Page 332:
“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon.  It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8. on Page 359:
“… eyes as objects which manifest the look.  The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9. on Page 392:
“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.  on Page 502:
“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”
11.  on Page 506:
“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.  on Page 661:
“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.  on Page 754:
“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself.  The pen and the pipe, the clothing, the desk, the house– are myself.  The totality of my possessions reflects the totality of my being.  I am what I have.  It is I myself which I touch in this cup, in this trinket.  This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

The image “http://www.log24.com/log/pix05/050527-CipraLogo.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050527-CIPRAview.jpg” cannot be displayed, because it contains errors.
 
From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

Sunday, February 20, 2005

Sunday February 20, 2005

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

Relativity Blues

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

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

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

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

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

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

A more thoughtful response would take into account the following:

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

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

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

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

From a review of Gilead by Jane Vandenburgh:  

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

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

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

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

A recent answer:

Modal Theology

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

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

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

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

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

Sunday, June 6, 2004

Sunday June 6, 2004

Filed under: General,Geometry — m759 @ 1:28 PM
Parallelisms

“I confess I do not believe in time.
I like to fold my magic carpet,
after use, in such a way
as to superimpose
one part of the pattern
upon another.”

(Nabokov, Speak, Memory)

From a review of On the Composition of Images, Signs & Ideas, by Giordano Bruno:

Proteus in the House of Mnemosyne (which is the fifth chapter of the Third Book) relies entirely on familiarity with Vergil’s Aeneid (even when the text shifts from verse to prose). The statement, “Proteus is, absolutely, that one and the same subject matter which is transformable into all images and resemblances, by means of which we can immediately and continually constitute order, resume and explain everything,” reads less clear than the immediate analogy, “Just as from one and the same wax we awaken all shapes and images of sensate things, which become thereafter the signs of all things that are intelligible.”

From an interview with Vladimir Nabokov published in Wisconsin Studies in Contemporary Literature, vol. VIII, no. 2, Spring 1967:

When I was your student, you never mentioned the  Homeric parallels in discussing Joyce’s Ulysses  But you did supply “special information” in introducing many of the masterpieces: a map of Dublin for Ulysses….  Would you be able to suggest some equivalent for your own readers?

Joyce himself very soon realized with dismay that the harping on those essentially easy and vulgar “Homeric parallelisms” would only distract one’s attention from the real beauty of his book. He soon dropped these pretentious chapter titles which already were “explaining” the book to non-readers.  In my lectures I tried to give factual data only. A map of three country estates with a winding river and a figure of the butterfly Parnassius mnemosyne for a cartographic cherub will be the endpaper in my revised edition of Speak, Memory.

For more on Joyce and Proteus,
see the May 27 entry
Ineluctable.

Thursday, May 20, 2004

Thursday May 20, 2004

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

Parable

“A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’), a term used to designate the geometric application we call a ‘parabola.’….  The basic parables are extended similes or metaphors.”

http://religion.rutgers.edu/nt/
    primer/parable.html

“If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals.  Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind.”

Sharon Begley, “The Puzzle of Genius,” Newsweek magazine, June 28, 1993, p. 50

“The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem ‘Pied Beauty’ has to do with ‘creation.’ “

Speaking in Parables, Ch. 2, by Sallie McFague

“The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels….  According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ‘see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”

— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

For further details, see

Speaking in Parables:
A Study in Metaphor and Theology

by Sallie McFague

Fortress Press, Philadelphia, 1975

Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7

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

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

For metaphor and algebra combined, see

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

Tuesday, September 16, 2003

Tuesday September 16, 2003

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

The Form, the Pattern

"…the sort of organization that Eliot later called musical, in his lecture 'The Music of Poetry', delivered in 1942, just as he was completing Four Quartets: 'The use of recurrent themes is as natural to poetry as to music,' Eliot says:

There are possibilities for verse which bear some analogy to the development of a theme by different groups of instruments [‘different voices’, we might say]; there are possibilities of transitions in a poem comparable to the different movements of a symphony or a quartet; there are possibilities of contrapuntal arrangement of subject-matter."

— Louis L. Martz, from
"Origins of Form in Four Quartets,"
in Words in Time: New Essays on Eliot’s Four Quartets, ed. Edward Lobb, University of Michigan Press, 1993

"…  Only by the form, the pattern,     
Can words or music reach
The stillness…."

— T. S. Eliot,
Four Quartets

Four Quartets

For a discussion of the above
form, or pattern, click here.

Monday, December 16, 2002

Monday December 16, 2002

Filed under: General — m759 @ 2:22 AM

Beethoven’s Birthday

“Ludwig van Beethoven’s String Quartet in A Minor, Opus 132, is one of the transcendent masterworks of the Western classical tradition. It is built around its luminous third movement, titled ‘Holy song of thanksgiving by one recovering from an illness.’

In this third movement, the aging Beethoven speaks, clearly and distinctly, in a voice seemingly meant both for all the world and for each individual who listens to it. The music, written in the ancient Lydian mode, is slow and grave and somehow both a struggle and a celebration at the same time.

This is music written by a supreme master at the height of his art, saying that through all illness, tribulation and sorrow there is a strength, there is a light, there is a hope.”

—  Andrew Lindemann Malone

“Eliot’s final poetic achievement—and, for many, his greatest—is the set of four poems published together in 1943 as Four Quartets…. Structurally—though the analogy is a loose one—Eliot modeled the Quartets on the late string quartets of Beethoven, especially… the A Minor Quartet; as early as 1931 he had written the poet Stephen Spender, ‘I have the A Minor Quartet on the gramophone, and I find it quite inexhaustible to study. There is a sort of heavenly or at least more than human gaiety about some of his later things which one imagines might come to oneself as the fruit of reconciliation and relief after immense suffering; I should like to get something of that into verse before I die.'”

— Anonymous author at a
Longman Publishers website

“Each of the late quartets has a unique structure, and the structure of the Quartet in A Minor is one of the most striking of all. Its five movements form an arch. At the center is a stunning slow movement that lasts nearly half the length of the entire quartet…

The third movement (Molto adagio) has a remarkable heading: in the score Beethoven titles it ‘Hymn of Thanksgiving to the Godhead from an Invalid,’ a clear reflection of the illness he had just come through. This is a variation movement, and Beethoven lays out the slow opening section, full of heartfelt music. But suddenly the music switches to D major and leaps ahead brightly; Beethoven marks this section ‘Feeling New Strength.’ These two sections alternate through this movement (the form is A-B-A-B-A), and the opening section is so varied on each reappearance that it seems to take on an entirely different character each time: each section is distinct, and each is moving in its own way (Beethoven marks the third ‘With the greatest feeling’). This movement has seemed to many listeners the greatest music Beethoven ever wrote. and perhaps the problem of all who try to write about this music is precisely that it cannot be described in words and should be experienced simply as music.”

—  Eric Bromberger,
Borromeo Quartet program notes 

In accordance with these passages, here is a web page with excellent transcriptions for piano by Steven Edwards of Beethoven’s late quartets:

The 16 String Quartets.

Our site music for today, Beethoven’s String Quartet No. 15 in A Minor, Opus 132, Movement 3 (1825), is taken from this web page.

Friday, August 30, 2002

Friday August 30, 2002

Filed under: General,Geometry — Tags: , — m759 @ 2:30 AM

For Mary Shelley, on her birthday: A Chain of Links The creator of Frankenstein might appreciate the following chain of thought. Lucifer.com Lucifer Media Corporation Lucifer Media Sites The Extropy Institute: International Transhumanist Solutions Why Super-Human Intelligence Would Be Equivalent To Precognition, by Marc Geddes:

"Consider the geometry of multiple dimensions as an analogy for mental abilities… …if there is a 4th dimension of intelligence, to us ordinary humans stuck with 3 dimensional reasoning, this 4th dimension would be indistinguishable from precognition. Post-humans would appear to us ordinary humans as beings which could predict the future in ways which would be inexplicable to us. We should label post-humans as 'Pre-Cogs.'

In the Steven Speilberg [sic]  film Minority Report, we encounter genetically engineered humans with precisely the abilities described above."

Internet Movie Database page on "Minority Report"

IMDb page on "Minority Report" author Philip K. Dick

IMDb biography of Philip K. Dick, where our chain of links ends.  Here Dick says that

"The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words."

On the other hand, Dick also says here that

"Reality is that which, when you stop believing in it, doesn't go away."

These two quotations summarize, on the one hand, the cynical, relativistic nominalism of the postmodernists and, on the other hand, the hard-nosed realism of the Platonists.

What does all this have to do with "the geometry of multiple dimensions"?

Consider the famous story for adolescents, A Wrinkle in Time, by Madeleine L'Engle.   The author, a well-meaning Christian, tries, like all storytellers,  to control her readers by controlling the meaning of words.   The key word in this book is "tesseract," a term from multi-dimensional geometry.   She insists that a tesseract has mystic properties and cannot be visualized.  She is wrong (at least about the visualizing).

See The Tesseract: A look into 4-dimensional space, by Harry J. Smith.

See also the many revealing comments in Harry J. Smith's Guestbook.

One of Smith's guests remarks, apropos of Smith's comments on St. Joseph, that he has his own connection with St. Augustine.

For a adult-level discussion of Augustine, time, eternity, and Platonism, see the website Time as a Psalm in St. Augustine, by A. M. Johnston.

See also the remark headlining Maureen Dowd's New York Times column of August 28, 2002, Saint Augustine's Day:

"I'm with Dick."

Whether the realist Dick or the nominalist Dick, she does not say.

As for precognition, see my series of journal notes below, which leads up to two intriguing errors in an Amazon.com site on the "Forbidden Planet" soundtrack.   The first two audio samples from this soundtrack are (wrongly) entitled "Birdland" and "Flamingo."  See also the West Wing episode rebroadcast on Wednesday, August 28, 2002,

The Black Vera Wang

C. J. Cregg (Allison Janney), who models a black Vera Wang dress in that episode, has the Secret Service codename Flamingo.

"…that woman in black She's a mystery She's everything a woman should be Woman in black got a hold on me"

(Foreigner 4 in my August 28 note below)

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