Log24

Friday, February 7, 2020

Correspondences

Filed under: General — Tags: , — m759 @ 1:05 PM

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)

Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.

See also Correspondences in this journal.

* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .

Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:

See as well the word correspondences  in 
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).

Sunday, July 17, 2016

Symmetries and Correspondences

Filed under: General — m759 @ 8:19 AM

Continued from  July 14, 2016


 

Symmetries and Correspondences in 1879 —

Cyparissos Stephanos
Sur les systèmes desmiques de trois tétraèdres
Bulletin des sciences mathématiques
et astronomiques 2e série
,
Tome 3, No. 1 (1879), pp. 424-456.
<http://www.numdam.org/item?id=BSMA_1879_2_3_1_424_1>
© Gauthier-Villars, 1879, tous droits réservés.


 

Symmetries and Correspondences in 1905 —

Thursday, July 14, 2016

Symmetries and Correspondences

Filed under: General,Geometry — Tags: — m759 @ 7:30 PM

The title is that of a large-scale British research project
in mathematics. On a more modest scale

"Hanks + Cube" in this journal —

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Block That Metaphor

Wednesday, January 6, 2016

Correspondences

Filed under: General — Tags: — m759 @ 10:06 PM

The above passage is from a Dec. 19, 2015, post,
Nunc Stans , on the death of New York Philharmonic
music director emeritus Kurt Masur.

See also a Log24 search for the word "Correspondences."

Monday, October 6, 2014

Mysterious Correspondences

Filed under: General,Geometry — m759 @ 9:36 AM

(Continued from Beautiful Mathematics, Dec. 14, 2013)

“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of 
a book review in the Dec. 5, 2013, issue
of 
The New York Review of Books

One such correspondence:

For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”

See as well posts from that same March 9 now tagged “Story Creep.”

Tuesday, May 7, 2013

Strange Correspondences

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

For Jerusalem Day

Strange Correspondences :

"There are interesting correspondences between
Jewish Kabbala, Torah, and Talmud, and
Chinese Buddhism and Taoism…."

Tony Smith

See also Chinese Checkers in this journal.

Saturday, August 6, 2011

Correspondences

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

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.

 

For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

Monday, February 17, 2020

RIP Charles Portis

Filed under: General — Tags: , — m759 @ 4:04 PM

     See also "True Grid " in this  journal.

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that 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, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
– The Club Dumas , 1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
– Robert Plant, 1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal— or, according to Krauss,
a "staircase" to the universal— is the ninefold square:

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

"This is the garden of Apollo, the field of Reason…."
– John Outram, architect    

See as well . . .

Tuesday, November 27, 2018

Fe

Filed under: General — m759 @ 10:38 AM

http://www.log24.com/log/pix18/180808-Cosima_drank_the_purple_kool-aid-500w.jpg

Thursday, October 6, 2016

A Labyrinth for Octavio

Filed under: General — m759 @ 7:00 PM

The title refers to the previous post.

From Middlemarch  (1871-2), by George Eliot, Ch. III —

"Dorothea by this time had looked deep into the ungauged reservoir of Mr. Casaubon's mind, seeing reflected there in vague labyrinthine extension every quality she herself brought; had opened much of her own experience to him, and had understood from him the scope of his great work, also of attractively labyrinthine extent. For he had been as instructive as Milton's 'affable archangel;' and with something of the archangelic manner he told her how he had undertaken to show (what indeed had been attempted before, but not with that thoroughness, justice of comparison, and effectiveness of arrangement at which Mr. Casaubon aimed) that all the mythical systems or erratic mythical fragments in the world were corruptions of a tradition originally revealed. Having once mastered the true position and taken a firm footing there, the vast field of mythical constructions became intelligible, nay, luminous with the reflected light of correspondences. But to gather in this great harvest of truth was no light or speedy work."

See also the term correspondence  in this journal.

Wednesday, October 5, 2016

Sources

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

From a Google image search yesterday

Sources (left to right, top to bottom) —

Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)

Thursday, October 22, 2015

Objective Quality

Filed under: General,Geometry — Tags: — m759 @ 2:26 AM

Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:

A more interesting account of these 35 structures:

"It is commonly known that there is a bijection between
the 35 unordered triples of a 7-set [i.e., the 35 partitions
of an 8-set into two 4-sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.

For some context, see Eightfold Geometry by Steven H. Cullinane.

Monday, October 6, 2014

Reviews

Filed under: General — m759 @ 11:00 AM

From the MacTutor biography of Otto Neugebauer:

“… two projects which would be among the most important
contributions anyone has made to mathematics. He persuaded
Springer-Verlag to publish a journal reviewing all mathematical
publications, which would complement their reviewing journals
in other topics. In 1931 the first issue of 
Zentralblatt für Matematik
appeared, edited by Neugebauer.” [Mathematical Reviews  was
the other project.]

Neugebauer appeared in Sunday morning’s post In Nomine Patris .

A review from Zentralblatt  appeared in the Story Creep link from
this morning’s post Mysterious Correspondences.

Monday, September 15, 2014

A Seventh Seal

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

This post was suggested by the two previous posts, Sermon and Structure.

IMAGE- Epigraph to Ch. 7 of Cameron's 'Parallelisms of Complete Designs'- '...fiddle with pentagrams...' from 'Four Quartets'

Vide  below the final paragraph— in Chapter 7— of Cameron’s Parallelisms ,
as well as Baudelaire in the post Correspondences :

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, “Correspondances “

A related image search (click to enlarge):

Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links.  See also a post of
​Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare  the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

Sunday, July 28, 2013

Sermon

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

(Simplicity continued)

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

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

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

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

A more sophisticated geometrical figure—

Galois-geometry key to
Desargues' theorem:

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

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

Friday, May 3, 2013

Structure

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

For the Church of St. Frank:

See Strange Correspondences and Eightfold Geometry.

Correspondences , by Steven H. Cullinane, August 6, 2011

The rest is the madness of art.”

Friday, January 20, 2012

The Nothing That Is

Filed under: General — m759 @ 9:00 PM

"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of  its imaginative correspondences,
has become 'the nothing that is.'"

Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought
.
New Brunswick: Rutgers UP, 1958.

 

IMAGE- The Ninefold Square at Ninevine.net

Monday, September 26, 2011

Inner and Outer

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

For T.S. Eliot's Birthday

Last night's post "Transformation" was suggested in part
by the title of a Sunday New York Times  article on
George Harrison, "Within Him, Without Him," and by
the song title "Within You Without You" in the post
Death and the Apple Tree.

Related material— "Hamlet's Transformation"—

Hamlet, 2.2:

Something have you heard
Of Hamlet’s transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was….”

A transformation:

The image “http://www.log24.com/theory/images/DTinscapes4-Trans.gif” cannot be displayed, because it contains errors.

Click on picture for details.

See also, from this year's Feast of the Transfiguration,
Correspondences and Happy Web Day.

For those who prefer the paganism of Yeats to
the Christianity of Eliot, there is the sequel to
"Death and the Apple Tree," "Dancers and the Dance."

Friday, August 12, 2011

Into the Woods

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

"What’s best about us, I hope, is that we teach them
the ‘forest of symbols,’ to borrow deliberately from
a poem called ‘Correspondences,’ by Baudelaire."

The late Stanley Bosworth, founding headmaster
    of St. Ann's School in Brooklyn

Bosworth died Sunday.

Related material—

  1. Saturday's Correspondences,
  2. Sunday's   Coordinated Steps, and
  3. Monday's  Organizing the Mine Workers.

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: General,Geometry — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Organizing the Mine Workers

Filed under: General — m759 @ 12:24 PM

From this journal on Saturday, August 6, 2011

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances " (in The Flowers of Evil )

From the New York Times  philosophy column "The Stone" earlier that day

"… a magnificent and colorful parade of disorganized and rhapsodic thoughts"

— Baudelaire

From Uncle Walt— (See yesterday's "Coordinated Steps")—

http://www.log24.com/log/pix11B/110808-DwarfsParade500w.jpg

For a better organized, less rhapsodic parade, see Saturday's Correspondences.

Saturday, August 6, 2011

Happy Web Day

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

Today the World Wide Web turns 20.

http://www.log24.com/log/pix11B/110806-EightfoldGeometrySearch.jpg

See also Galois Memorial and Correspondences.

Saturday, January 8, 2011

True Grid (continued)

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

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

Sunday December 10, 2006

 

 m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that 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, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

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

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

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

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Tuesday, August 18, 2009

Tuesday August 18, 2009

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

Prima Materia

(Background: Art Humor: Sein Feld (March 11, 2009) and Ides of March Sermon, 2009)

From Cardinal Manning’s review of Kirkman’s Philosophy Without Assumptions

“And here I must confess… that between something and nothing I can find no intermediate except potentia, which does not mean force but possibility.”

— Contemporary Review, Vol. 28 (June-November, 1876), page 1017

Furthermore….

Cardinal Manning, Contemporary Review, Vol. 28, pages 1026-1027:

The following will be, I believe, a correct statement of the Scholastic teaching:–

1. By strict process of reason we demonstrate a First Existence, a First Cause, a First Mover; and that this Existence, Cause, and Mover is Intelligence and Power.

2. This Power is eternal, and from all eternity has been in its fullest amplitude; nothing in it is latent, dormant, or in germ: but its whole existence is in actu, that is, in actual perfection, and in complete expansion or actuality. In other words God is Actus Purus, in whose being nothing is potential, in potentia, but in Him all things potentially exist.

3. In the power of God, therefore, exists the original matter (prima materia) of all things; but that prima materia is pura potentia, a nihilo distincta, a mere potentiality or possibility; nevertheless, it is not a nothing, but a possible existence. When it is said that the prima materia of all things exists in the power of God, it does not mean that it is of the existence of God, which would involve Pantheism, but that its actual existence is possible.

4. Of things possible by the power of God, some come into actual existence, and their existence is determined by the impression of a form upon this materia prima. The form is the first act which determines the existence and the species of each, and this act is wrought by the will and power of God. By this union of form with the materia prima, the materia secunda or the materia signata is constituted.

5. This form is called forma substantialis because it determines the being of each existence, and is the root of all its properties and the cause of all its operations.

6. And yet the materia prima has no actual existence before the form is impressed. They come into existence simultaneously;

[p. 1027 begins]

as the voice and articulation, to use St. Augustine’s illustration, are simultaneous in speech.

7. In all existing things there are, therefore, two principles; the one active, which is the form– the other passive, which is the matter; but when united, they have a unity which determines the existence of the species. The form is that by which each is what it is.

8. It is the form that gives to each its unity of cohesion, its law, and its specific nature.*

When, therefore, we are asked whether matter exists or no, we answer, It is as certain that matter exists as that form exists; but all the phenomena which fall under sense prove the existence of the unity, cohesion, species, that is, of the form of each, and this is a proof that what was once in mere possibility is now in actual existence. It is, and that is both form and matter.

When we are further asked what is matter, we answer readily, It is not God, nor the substance of God; nor the presence of God arrayed in phenomena; nor the uncreated will of God veiled in a world of illusions, deluding us with shadows into the belief of substance: much less is it catter [pejorative term in the book under review], and still less is it nothing. It is a reality, the physical kind or nature of which is as unknown in its quiddity or quality as its existence is certainly known to the reason of man.

* “… its specific nature”
        (Click to enlarge) —

Footnote by Cardinal Manning on Aquinas
The Catholic physics expounded by Cardinal Manning above is the physics of Aristotle.

For a more modern treatment of these topics, see Werner Heisenberg’s Physics and Philosophy. For instance:

“The probability wave of Bohr, Kramers, Slater, however, meant… a tendency for something. It was a quantitative version of the old concept of ‘potentia’ in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.”

Compare to Cardinal Manning’s statement above:

“… between something and nothing I can find no intermediate except potentia…”

To the mathematician, the cardinal’s statement suggests the set of real numbers between 1 and 0, inclusive, by which probabilities are measured. Mappings of purely physical events to this set of numbers are perhaps better described by applied mathematicians and physicists than by philosophers, theologians, or storytellers. (Cf. Voltaire’s mockery of possible-worlds philosophy and, more recently, The Onion‘s mockery of the fictional storyteller Fournier’s quantum flux. See also Mathematics and Narrative.)

Regarding events that are not purely physical– those that have meaning for mankind, and perhaps for God– events affecting conception, birth, life, and death– the remarks of applied mathematicians and physicists are often ignorant and obnoxious, and very often do more harm than good. For such meaningful events, the philosophers, theologians, and storytellers are better guides. See, for instance, the works of Jung and those of his school. Meaningful events sometimes (perhaps, to God, always) exhibit striking correspondences. For the study of such correspondences, the compact topological space [0, 1] discussed above is perhaps less helpful than the finite Galois field GF(64)– in its guise as the I Ching. Those who insist on dragging God into the picture may consult St. Augustine’s Day, 2006, and Hitler’s Still Point.

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:07 PM
Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

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

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

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler
 
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

 

Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today

 

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

"Pauli linked this symbolism
with the concept of automorphism."

The Innermost Kernel
 (previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic
(4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible."

"There is such a thing
as a tesseract."

Madeleine L'Engle 

Sunday, April 8, 2007

Sunday April 8, 2007

Filed under: General — Tags: — m759 @ 12:00 AM
Midnight in the Garden
continued from Sept. 30, 2004

Tonight this journal had two Xanga footprints from Italy….

At 11:34 PM ET a visitor from Italy viewed a page containing an entry from Jan. 8, 2005, Splendor of the Light, which offers the following quotation–

From an essay on Guy Davenport
 

"A disciple of Ezra Pound, he adapts to the short story the ideogrammatic method of The Cantos, where a grammar of images, emblems, and symbols replaces that of logical sequence. This grammar allows for the grafting of particulars into a congeries of implied relation without subordination. In contrast to postmodernists, Davenport does not omit causal connection and linear narrative continuity for the sake of an aleatory play of signification but in order to intimate by combinational logic kinships and correspondences among eras, ideas and forces."

— "When Novelists Become Cubists: The Prose Ideograms of Guy Davenport," by Andre Furlani

The visitor from Italy may, of course, have instead intended to view one of the four earlier entries on the page.  In particular, the visitor may have seen

The Star
of Venus

"He looked at the fading light
in the western sky and saw Mercury,
or perhaps it was Venus,
gleaming at him as the evening star.
Darkness and light,
the old man thought.
It is what every hero legend is about.
The darkness which is more than death,
the light which is love, like our friend
Venus here, or perhaps this star is
Mercury, the messenger of Olympus,
the bringer of hope."

Roderick MacLeish, Prince Ombra.

At 11:38 PM ET, a visitor from Italy (very likely the 11:34 visitor returning) viewed the five Log24 entries ending at 12:06 AM ET on Sept. 30, 2004. 

These entries included Midnight in the Garden and…

A Tune for Michaelmas

Mozart, K 265, midi

The entries on this second visited page also included some remarks on Dante, on time, and on Malcolm Lowry's Under the Volcano that are relevant to Log24 entries earlier this week on Maundy Thursday and on Holy Saturday.

Here's wishing a happy Easter to Italy, to Francis Ford Coppola and Russell Crowe (see yesterday's entry), and to Steven Spielberg (see the Easter page of April 20, 2003).

The image “http://www.log24.com/log/pix07/070408-Prayer.jpg” cannot be displayed, because it contains errors.

Image courtesy of
Hollywood Jesus:

When you wish
upon a star…

Sunday, November 26, 2006

Sunday November 26, 2006

Filed under: General — m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that 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, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
The Club Dumas,1993

"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
Robert Plant,1971

The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.

An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:

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

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect    

For more on the field
of reason, see
Log24, Oct. 9, 2006.

A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).

Unreason is, of course,
more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"

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

Wednesday, August 2, 2006

Wednesday August 2, 2006

Filed under: General — m759 @ 2:00 AM
Final Arrangements,
continued

Ontology Alignment is
the process of determining
correspondences between concepts.”

The image “http://www.log24.com/log/pix06A/060802-Deaths.jpg” cannot be displayed, because it contains errors.

Online New York Times today

The image “http://www.log24.com/log/pix05/050326-Garden.jpg” cannot be displayed, because it contains errors.

“With a little effort,
anything can be shown to
connect with anything else:
existence is infinitely
cross-referenced.”

— Opening sentence of
Martha Cooley’s The Archivist

“Frere Jacques, Cuernavaca,
ach du lieber August.”

— John O’Hara, Hope of Heaven, 1938

And now I was beginning to surmise:
Here was the library of Paradise.

Hermann Hesse, Magister Ludi 

(For Hesse in another context,
see the Log24 entries of
  Nov. 4-6, 2003.)

Wednesday, July 26, 2006

Wednesday July 26, 2006

Filed under: General — Tags: — m759 @ 1:44 PM
Partitions,
continued

"Mistakes are inevitable and may be either in missing a true signal or in thinking there is a signal when there is not. I am suggesting that believers in the paranormal (called 'sheep' in psychological parlance) are more likely to make the latter kind of error than are disbelievers (called 'goats')."

— "Psychic Experiences:
     Psychic Illusions,"
     by Susan Blackmore,
     Skeptical Inquirer, 1992

For Harvard mathematician
Frederick Mosteller,
dead on Sunday, July 23, 2006:
 
"… a drama built out of nothing
but numbers and imagination"

— Freeman Dyson, quoted in Log24
on the day Mosteller died

From Log24 on
Mosteller's last birthday,
December 24, 2005:

The Club Dumas

by Arturo Perez-Reverte

One by one, he tore the engravings from the book, until he had all nine.  He looked at them closely.  "It's a pity you can't follow me where I'm going.  As the fourth engraving states, fate is not the same for all."

"Where do you believe you're going?"

Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

"To meet someone" was his enigmatic answer. "To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso."

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two)

Carl Gustav Jung,   
born on this date

Today's other birthday:
Mick Jagger

"Pleased to meet you,
hope you guess my name."

Saturday, December 24, 2005

Saturday December 24, 2005

Filed under: General — m759 @ 12:08 AM
The Stone of Power

“Others say it is a stone that posseses mysterious powers…. often depicted as a dazzling light.  It’s a symbol representing power, a source of immense energy.  It nourishes, heals, wounds, blinds, strikes down…. Some have thought of it as the philosopher’s stone of the alchemists….”

Foucault’s Pendulum 
by Umberto Eco,
Professor of Semiotics

The Club Dumas

by Arturo Perez-Reverte

(Paperback, pages 346-347):

One by one, he tore the engravings from the book, until he had all nine.  He looked at them closely.  “It’s a pity you can’t follow me where I’m going.  As the fourth engraving states, fate is not the same for all.”

“Where do you believe you’re going?”

Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

“To meet someone” was his enigmatic answer. “To search for the stone that the Great Architect rejected, the philosopher’s stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.”

“Only gradually did I discover
what the mandala really is:
‘Formation, Transformation,
Eternal Mind’s eternal recreation'”
(Faust, Part Two)

Carl Gustav Jung  

Wednesday, December 21, 2005

Wednesday December 21, 2005

Filed under: General,Geometry — m759 @ 4:07 PM
For the feast of
St. Francis Scott Key Fitzgerald

The Diamond
as Big as
the Monster

From Fitzgerald’s The Diamond as Big as the Ritz:

    “Now,” said John eagerly, “turn out your pocket and let’s see what jewels you brought along. If you made a good selection we three ought to live comfortably all the rest of our lives.”
     Obediently Kismine put her hand in her pocket and tossed two handfuls of glittering stones before him.
    “Not so bad,” cried John, enthusiastically. “They aren’t very big, but– Hello!” His expression changed as he held one of them up to the declining sun. “Why, these aren’t diamonds! There’s something the matter!”
    “By golly!” exclaimed Kismine, with a startled look. “What an idiot I am!”
    “Why, these are rhinestones!” cried John.

From The Hawkline Monster, by Richard Brautigan:
 
    “What are we going to do now?” Susan Hawkline said, surveying the lake that had once been their house.
    Cameron counted the diamonds in his hand.  There were thirty-five diamonds and they were all that was left of the Hawkline Monster.
    “We’ll think of something,” Cameron said.

Related material:

“A disciple of Ezra Pound, he adapts to the short story the ideogrammatic method of The Cantos, where a grammar of images, emblems, and symbols replaces that of logical sequence. This grammar allows for the grafting of particulars into a congeries of implied relation without subordination. In contrast to postmodernists, Davenport does not omit causal connection and linear narrative continuity for the sake of an aleatory play of signification but in order to intimate by combinational logic kinships and correspondences among eras, ideas and forces.”

When Novelists Become Cubists:
    The Prose Ideograms of Guy Davenport,
    by Andre Furlani

“T.S. Eliot’s experiments in ideogrammatic method are equally germane to Davenport, who shares with the poet an avant-garde aesthetic and a conservative temperament.  Davenport’s text reverberates with echoes of Four Quartets.”

Andre Furlani

“At the still point,
  there the dance is.”

—  T. S. Eliot, Four Quartets,
quoted in the epigraph to
the chapter on automorphism groups
in Parallelisms of Complete Designs,
by Peter J. Cameron,
published when Cameron was at
Merton College, Oxford.

“As Gatsby closed the door of
‘the Merton College Library’
I could have sworn I heard
the owl-eyed man
break into ghostly laughter.”

F. Scott Fitzgerald

Sunday, January 9, 2005

Sunday January 9, 2005

Filed under: General — m759 @ 3:10 PM

Light at Bologna

“Others say it is a stone that posseses mysterious powers…. often depicted as a dazzling light.  It’s a symbol representing power, a source of immense energy.  It nourishes, heals, wounds, blinds, strikes down…. Some have thought of it as the philosopher’s stone of the alchemists….”

Foucault’s Pendulum
by Umberto Eco,
Professor of Semiotics at
 Europe’s oldest university,
 the University of Bologna.

The Club Dumas

by Arturo Perez-Reverte

(Paperback, pages 346-347):

One by one, he tore the engravings from the book, until he had all nine.  He looked at them closely.  “It’s a pity you can’t follow me where I’m going.  As the fourth engraving states, fate is not the same for all.”

“Where do you believe you’re going?”

Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

“To meet someone” was his enigmatic answer. “To search for the stone that the Great Architect rejected, the philosopher’s stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso. From Faust’s black dog to the false angel of light who tried to break down Saint Anthony’s resistance.  But most of all, stupidity bores him, and he hates monotony….”

Eclogues: Eight Stories

by Guy Davenport

Johns Hopkins paperback, 1993, page 127 —

Lo Splendore della Luce a Bologna, VI:

“In 1603, at Monte Paderno, outside Bologna, an alchemist (by day a cobbler) named Vicenzo Cascariolo discovered the Philosopher’s Stone, catalyst in the transformation of base metals into gold, focus of the imagination, talisman for abstruse thought.  Silver in some lights, white in others, it glowed blue in darkness, awesome to behold.”


The Discovery of Luminescence:

The Bolognian Stone

The image “http://www.log24.com/log/pix05/050109-Bologna.jpg” cannot be displayed, because it contains errors.
Bologna, 16th Century

“For the University of Bologna hosting an International Conference on Bioluminescence and Chemiluminescence has a very special significance. Indeed, it is in our fair City that modern scientific research on these phenomena has its earliest roots….

‘After submitting the stone
to much preparation, it was not
the Pluto of Aristophanes
that resulted; instead, it was
the Luciferous Stone’ ”

From one of the best books
of the 20th century:

The Hawkline Monster

by Richard Brautigan

The image “http://www.log24.com/log/pix05/050109-Hawkline.jpg” cannot be displayed, because it contains errors.
“The Chemicals that resided in the jar were a combination of hundreds of things from all over the world.  Some of The Chemicals were ancient and very difficult to obtain.  There were a few drops of something from an Egyptian pyramid dating from the year 3000 B.C.

There were distillates from the jungles of South America and drops of things from plants that grew near the snowline in the Himalayas.

Ancient China, Rome and Greece had contributed things, too, that had found their way into the jar.  Witchcraft and modern science, the latest of discoveries, had also contributed to the contents of the jar.  There was even something that was reputed to have come all the way from Atlantis….

… they did not know that the monster was an illusion created by a mutated light in The Chemicals. a light that had the power to work its will upon mind and matter and change the very nature of reality to fit its mischievous mind.”

Saturday, January 8, 2005

Saturday January 8, 2005

Filed under: General — m759 @ 7:26 PM
Splendor of the Light

The Beginning of a Story
by Guy Davenport

Lo Splendore della Luce a Bologna

“The locomotive bringing a trainload of philosophers to Bologna hissed and ground to a standstill in the long Appenine dusk to have its headlamps lit and to be dressed in the standards of the city and the university.”

Eclogues, by Guy Davenport
(Johns Hopkins paperbacks
   ed. edition, 1993, page 125)

Related material:

The train wreck at 12:50 pm local time (6:50 AM EST) Friday, Jan. 7, 2005, 25 miles north of Bologna.

A northbound freight train collided with a passenger train traveling south from Verona to Bologna.

From an essay on Davenport I found Friday morning, well before I learned on Friday afternoon (Eastern Standard Time) of the train wreck:

“A disciple of Ezra Pound, he adapts to the short story the ideogrammatic method of The Cantos, where a grammar of images, emblems, and symbols replaces that of logical sequence. This grammar allows for the grafting of particulars into a congeries of implied relation without subordination. In contrast to postmodernists, Davenport does not omit causal connection and linear narrative continuity for the sake of an aleatory play of signification but in order to intimate by combinational logic kinships and correspondences among eras, ideas and forces.”

— “When Novelists Become Cubists:
    The Prose Ideograms of Guy Davenport,”
by Andre Furlani

See also
Friday’s Log24 entries and
Davenport’s Express.

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