Sunday, January 29, 2006

Sunday January 29, 2006

Filed under: General — m759 @ 10:00 AM
New Haven

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The eye’s plain version is a thing apart,
The vulgate of experience.  Of this,
A few words, an and yet, and yet, and yet–

As part of the never-ending meditation,
Part of the question that is a giant himself:
Of what is this house composed if not of the sun,

These houses, these difficult objects, dilapidate
Appearances of what appearances,
Words, lines, not meanings, not communications,

Dark things without a double, after all,
Unless a second giant kills the first–
A recent imagining of reality,

Much like a new resemblance of the sun,
Down-pouring, up-springing and inevitable,
A larger poem for a larger audience,

As if the crude collops came together as one,
A mythological form, a festival sphere,
A great bosom, beard and being, alive with age.

— Wallace Stevens, opening lines of
    “An Ordinary Evening in New Haven”

Sunday January 29, 2006

Filed under: General — m759 @ 9:00 AM

In the punctual centre of all circles white
     Stands truly….

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… and Bloom with his vast accumulation
Stands and regards and repeats the primitive lines.

— Wallace Stevens,
“From the Packet of Anacharsis”

Related material:
Balanchine’s Birthday.

Sunday January 29, 2006

Filed under: General — m759 @ 8:00 AM
Sunday Morning

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Today is the Chinese New Year, 4074.

Saturday, January 28, 2006

Saturday January 28, 2006

Filed under: General — m759 @ 12:48 PM

Memorial of
St. Thomas Aquinas:

Joyce and Aquinas
by William T. Noon, S. J.

St. Thomas Aquinas,
by G. K. Chesterton

Log24, Epiphany 2006

Friday, January 27, 2006

Friday January 27, 2006

Filed under: General — m759 @ 8:25 AM

Mozart, 2006

Mozart, 1935

Poet, be seated at the piano.
Play the present, its hoo-hoo-hoo,
Its shoo-shoo-shoo, its ric-a-nic,
Its envious cachinnation.

If they throw stones upon the roof
While you practice arpeggios,
It is because they carry down the stairs
A body in rags.
Be seated at the piano.

That lucid souvenir of the past,
The divertimento;
That airy dream of the future,
The unclouded concerto . . .
The snow is falling.
Strike the piercing chord.

Be thou the voice,
Not you. Be thou, be thou
The voice of angry fear,
The voice of this besieging pain.

Be thou that wintry sound
As of the great wind howling,
By which sorrow is released,
Dismissed, absolved
In a starry placating.

We may return to Mozart.
He was young, and we, we are old.
The snow is falling
And the streets are full of cries.
Be seated, thou.

— Wallace Stevens, Ideas of Order (1936)

From the center:

“‘Mozart, 1935’ immediately discloses a will to counter complaints of pure poetry, to refute that charge heard regularly from Stevens’s critics, to find ‘his particular celebration out of tune today’ on his own if necessary; and, in short, to meet the communist [poet and critic Willard] Maas’s ‘respect for his magnificent rhetoric’ at least halfway across from right to left.”

— Alan Filreis, Modernism from Right to Left: Wallace Stevens, the Thirties, and Literary Radicalism (Cambridge: Cambridge University Press, 1994), p. 211

From the left:

Norman Lebrecht on this year’s celebration of the 250th anniversary of Mozart’s birth (January 27, 1756):

   “… Mozart, it is safe to say, failed to take music one step forward….
   … Mozart merely filled the space between staves with chords that he knew would gratify a pampered audience. He was a provider of easy listening, a progenitor of Muzak….
   …  He lacked the rage of justice that pushed Beethoven into isolation, or any urge to change the world. Mozart wrote a little night music for the ancien regime. He was not so much reactionary as regressive….
   … Little in such a mediocre life gives cause for celebration….
   … The bandwaggon of Mozart commemorations was invented by the Nazis in 1941….
   …  In this orgy of simple-mindedness, the concurrent centenary of Dmitri Shostakovich– a composer of true courage and historical significance– is being shunted to the sidelines, celebrated by the few.
    Mozart is a menace to musical progress, a relic of rituals that were losing relevance in his own time and are meaningless to ours. Beyond a superficial beauty and structural certainty, Mozart has nothing to give to mind or spirit in the 21st century. Let him rest. Ignore the commercial onslaught. Play the Leningrad Symphony. Listen to music that matters.”
The left seems little changed since 1935.

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

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The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

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See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

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Wednesday, January 25, 2006

Wednesday January 25, 2006

Filed under: General — m759 @ 4:00 PM
Born Today
and playing
with a full deck:

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Alicia Keys

“… it’s going to be
accomplished in steps,
this establishment
of the Talented in
  the scheme of things.”

— Anne McCaffrey, 
Radcliffe ’47,
To Ride Pegasus

(And born yesterday…
Neil “I am, I cried” Diamond)

Tuesday, January 24, 2006

Tuesday January 24, 2006

Filed under: General — Tags: — m759 @ 7:00 AM
for Michael Harris
(See previous entry.)

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Related material:
A classic book in a postmodern
(“free-floating signs”) cover —

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This is my Princeton Companion
to Mathematics
, from the days
when Princeton University Press
had higher scholarly standards.

Monday, January 23, 2006

Monday January 23, 2006

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

In Defense of Hilbert
(On His Birthday)

Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:

“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”

Harris adds in a footnote:

“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”

If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”

For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):

[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”

Some of Zach’s references:

[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].

[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990

From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:

“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”

[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….

[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.

[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.

From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:

“What Hilbert wants to secure is not the truth, but the consistency of the old analysis.  This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics.  In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one.  The statements become meaningless figures built up from signs.  Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess.  Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs.  One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess.  And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae.  By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game.  The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules.  Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color.  Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game.  Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge.  What is meant to be recognized is that a contradiction can never occur as an end formula of a proof.  Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration.  One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color.  In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game.  It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together.  Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”

Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:

“Hilbert’s program… received its share of criticism.  Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’  But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time?  I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous.  It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”

— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)

Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:

“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”

One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136.  The quotation in the English translation given there:

“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”

Corfield says of this quotation that he’d love to be told the original German.  He should consult the above references cited by Richard Zach.

For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.

Monday January 23, 2006

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

The Case

An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:

From The Hustler, by Walter Tevis:

The only light in the room was from the lamp over the couch where she was reading.
    He looked at her face.  She was very drunk.  Her eyes were swollen, pink at the corners.  "What's the book?" he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
    She blinked up at him, smiled sleepily, and said nothing.
    "What's the book?"  His voice had an edge now.
    "Oh," she said.  "It's Kierkegaard.  Soren Kierkegaard."  She pushed her legs out straight on the couch, stretching her feet.  Her skirt fell back a few inches from her knees.  He looked away.
    "What's that?" he said.
    "Well, I don't exactly know, myself."  Her voice was soft and thick.
    He turned his face away from her again, not knowing what he was angry with.  "What does that mean, you don't know, yourself?"
    She blinked at him.  "It means, Eddie, that I don't exactly know what the book is about.  Somebody told me to read it, once, and that's what I'm doing.  Reading it."
    He looked at her, tried to grin at her– the old, meaningless, automatic grin, the grin that made everybody like him– but he could not.  "That's great," he said, and it came out with more irritation than he had intended.
    She closed the book, tucked it beside her on the couch.  "I guess this isn't your night, Eddie.  Why don't we have a drink?"
    "No."  He did not like that, did not want her being nice to him, forgiving.  Nor did he want a drink.
    Her smile, her drunk, amused smile, did not change.  "Then let's talk about something else," she said.  "What about that case you have?  What's in it?"  Her voice was not prying, only friendly.  "Pencils?"
    "That's it," he said.  "Pencils."
    She raised her eyebrows slightly.  Her voice seemed thick.  "What's in it, Eddie?"
    "Figure it out yourself."  He tossed the case on the couch.


Related material:

Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,

The Diamond of Possibility,

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the Baseball Almanac,

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and this morning's entry, "Natural Hustler."

Monday January 23, 2006

Filed under: General — m759 @ 3:57 AM

Natural Hustler (jpg, 283 KB)

Sunday, January 22, 2006

Sunday January 22, 2006

Filed under: General — Tags: — m759 @ 2:08 PM
On this date in 1938, Thornton Wilder's
"Our Town" premiered at the
McCarter Theatre, Princeton University.

Related material:

  St. Patrick's Day, 2005,
  St. Patrick's Day, 2003,
 and, for
Piper Laurie's birthday
(today) in 2003,

Through a Soda-Fountain
Mirror, Darkly

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Sunday January 22, 2006

Filed under: General — Tags: — m759 @ 7:00 AM
Fourstone Parable

Alms for Oblivion:

IMAGE- Yale Babylonian tablet 7289 and Father Time

In memory of Akkadian scholar
Erica Reiner, who died at 81 on
December 31, 2005.

"Erica combined a tough-minded commitment to intellectual excellence with a dry wit, charm, and a deep love of art, music, and literature.  Erica's passion for her work was legendary. She was someone who expected the very highest standards of scholarly rigor both in her own work, and in the efforts of others."

Gil Stein, director of the Oriental Institute at the University of Chicago

A Mass for Dr. Reiner
was scheduled for
Friday the 13th
  at the church of
"doubting Thomas"–
St. Thomas the Apostle
in Chicago.

Sunday January 22, 2006

Filed under: General — Tags: — m759 @ 4:07 AM
Mathematics and Narrative

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Saturday, January 21, 2006

Saturday January 21, 2006

Filed under: General — m759 @ 7:48 AM

Jews on Fiction

See Tony Kushner and E.L. Doctorow in today’s New York Times, Rebecca Goldstein’s talk from last summer’s Mykonos conference on mathematics and narrative, and Martin Buber on the Bible.

Friday, January 20, 2006

Friday January 20, 2006

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

"Wherefore let it hardly… be… thought that the prisoner… was at his best a onestone parable
for… pathetically few… cared… to doubt… the canonicity of his existence as a tesseract."

Finnegans Wake, page 100, abridged

"… we have forgotten that we were angels and painted ourselves into a corner
of resource extraction and commodification of ourselves."

— A discussion, in a draft of a paper (rtf) attributed to Josh Schultz, 
of the poem "Diamond" by Attila Jozsef

Commodification of
the name Cullinane:

See the logos at
a design firm with
the motto

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(Note the 4Cs theme.)

To adapt a phrase from
Finnegans Wake, the
"fourstone parable" below
is an attempt to
decommodify my name.

Fourstone Parable:

(See also yesterday's "Logos."
The "communicate" logo is taken from
an online library at Calvin College;
the "connect" logo is a commonly
available picture of a tesseract
(Coxeter, Regular Polytopes, p. 123),
and the other two logos
are more or less original.)

For a more elegant
four-diamond figure, see
Jung and the Imago Dei.

Thursday, January 19, 2006

Thursday January 19, 2006

Filed under: General — m759 @ 8:00 PM
In Memoriam
Wilson Pickett
The Distinguished Gentleman

                     VERA JOHNSON
I'm not sure I understand, Professor
Franklin -- you wrote your doctoral
thesis on my husband?
He was a great man, Mrs. Johnson.
He did so much for my people.
He... did?
Oh, yes. I'll never forget when he
said 'Welfare is a drug -- and you
gotta kick it cold turkey.' It
was... inspirational.
Really... well, I'm sure...
And I was actually in the audience
when he said, 'If you people would
just get off your dead asses and
look for work, this country might
be fit to live in again.' Powerful
It's very kind of you to say so.
And you're very kind to come all
the way from... where was it?
Wilson Pickett State Teachers College.
But I didn't just come to pay
respects, ma'am. I came because
your husband deserves an archive.
A place where scholars can study
his legacy. A storehouse for the
record of his remarkable career.
I see. So you want... his papers?
Oh, not just his papers, Mrs.
Johnson. Everything. Buttons,
posters, bumper stickers. All the
paraphernalia of his campaigns --
proof of his political genius. Now
I realize that you may have a
sentimental attachment to a few
Take 'em.

Thursday January 19, 2006

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


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Alvin Plantinga

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Harry Plantinga

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Thursday January 19, 2006

Filed under: General — m759 @ 7:48 AM
The Man Who Was Thursday:
An Introduction

“Wallace Stevens’s remarkable oeuvre is a quasi-spiritual quest for the supreme fiction, for a poetry that ‘must take the place / Of empty heaven and its hymns’ and thus help modern man find meaning in a godless world. The poet’s role, for Stevens, is that of high priest of the imagination: it is the poet who ‘gives to life the supreme fictions without which we are unable to conceive of it.’ ….
… Stevens’s hallmark ‘imagination-reality’ complex… is pursued almost obsessively in his poetry and prose of the 1940s. Parts of a World, published in 1942, and the poem-sequence of the same year, ‘Notes toward a Supreme Fiction’ (‘Notes’ was subsequently collected in Transport to Summer in 1947), comprise a prolonged meditation in a time of war on poetry and the poet’s role, in the face of what Stevens, in his essay ‘The Noble Rider and the Sound of Words,’ terms ‘the pressure of reality.’ Parts of a World is riven by its competing vocabularies. A discourse of desire, of process, of the poet’s contemplation of the mind in the act of finding what will suffice, is elaborated in ‘the never-resting mind’ of ‘The Poems of Our Climate’ and in ‘The Well Dressed Man with a Beard,’ in which ‘It can never be satisfied, the mind, never’ [occurs]. A very different idiom, that of the ‘hero’ or ‘major man,’ the figure of capable imagination, dominates and directs such poems as ‘Mrs Alfred Uruguay,’ ‘Asides on the Oboe’ and ‘Examination of the Hero in a Time of War,’ where

    Summer, jangling
         the savagest diamonds and
    Dressed in its
         azure-doubled crimsons,
    May truly bear
         its heroic fortunes
    For the large,
         the solitary figure.”

Lee M. Jenkins,
    University College Cork,
   “Wallace Stevens,”
    The Literary Encyclopedia,
    9 Dec., 2004.

For some related serious, but less solemn, remarks, click on the above date.

Thursday January 19, 2006

Filed under: General — m759 @ 6:23 AM
Plato and Shakespeare
at Breakfast


"Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living, to know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast. He is always expecting to see some truth that he has never seen before."

G. K. Chesterton, Orthodoxy

For Plato:

For Shakespeare:
Hopkins on Inscape.

For both:

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Click on the picture
for related remarks.

Tuesday, January 17, 2006

Tuesday January 17, 2006

Filed under: General — m759 @ 11:00 PM
Bang Splat

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School Book Depository



“Many people look at the Kennedy assassination as a turning point, when people started realizing and thinking and believing their government would lie to them and lie to them repeatedly,” said Gary Mack, curator of the Sixth Floor Museum at Dealey Plaza in Dallas.

AP, Dallas, Nov. 21, 2003   

Better late than never.

Tuesday January 17, 2006

Filed under: General — m759 @ 12:00 PM
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BBC News Jan. 17

Related material:
Log24  Sept. 27 and
Sept. 28, 2005,
as well as
The Harvard Crimson,
Jan. 13, 2006:
“President was resolute–
‘This is bullshit'”

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Monday, January 16, 2006

Monday January 16, 2006

Filed under: General — m759 @ 4:00 AM

Mathematics and Narrative

Rebecca Goldstein, Mathematics and the Character of Tragedy:

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


Sunday, January 15, 2006

Sunday January 15, 2006

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


My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S6] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S6 contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S6 by a subgroup acting triply transitively on the 6 letters as PGL2(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

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

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

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

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

Saturday, January 14, 2006

Saturday January 14, 2006

Filed under: General — m759 @ 4:07 AM
Diamond Jubilance
(See previous entry.)

“A (very brief!) lit search reveals very little on the intersection between probability theory and modal logic…. probability and modality are such big topics one would think there’d be something on their intersection, and I don’t think the way I’ve framed the problem is entirely idiosyncratic.”

— Brian Weatherson, Associate Professor of Philosophy, Cornell University, May 11, 2004

Here, on the other hand, is a way of framing the problem that is entirely idiosyncratic:

On this date:

In 1970, William Feller died.
In 1978, Kurt Gödel died.
In 1898, the Rev. Deacon Charles Lutwidge Dodgson died.

Related material:
Log24, Jan. 14, 2003, and

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

Modal Theology.

Friday, January 13, 2006

Friday January 13, 2006

Filed under: General — m759 @ 12:00 PM

Beyond the Fire

“Who Needs a White Cube These Days?”
Headline in today’s New York Times

That Nature is a Heraclitean Fire…
— Poem title, Gerard Manley Hopkins

“… Sleep realized
Was the whiteness that is the ultimate intellect,
A diamond jubilance beyond the fire,

That gives its power to the wild-ringed eye.”

— Wallace Stevens,
   “The Owl in the Sarcophagus”
III 13-16,
    from The Auroras of Autumn, 1950

Related material:
The five entries ending on Christmas, 2005.

Wednesday, January 11, 2006

Wednesday January 11, 2006

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

Time in the Rock

"a world of selves trying to remember the self
before the idea of self is lost–

Walk with me world, upon my right hand walk,
speak to me Babel, that I may strive to assemble
of all these syllables a single word
before the purpose of speech is gone."

— Conrad Aiken, "Prelude" (1932),
    later part of "Time in the Rock,
    or Preludes to Definition, XIX" (1936),
    in Selected Poems, Oxford U. Press
    paperback, 2003, page 156

"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.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,

The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined

By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep."

— Wallace Stevens in The Rock (1954)

"Poetry is an illumination of a surface,
  the movement of a self in the rock."
— Wallace Stevens, introduction to
    The Necessary Angel, 1951

Related material:
Jung's Imago and Solomon's Cube.


The following may help illuminate the previous entry:

"I want, as a man of the imagination, to write poetry with all the power of a monster equal in strength to that of the monster about whom I write.  I want man's imagination to be completely adequate in the face of reality."

— Wallace Stevens, 1953 (Letters 790)

The "monster" of the previous entry is of course not Reese Witherspoon, but rather Vox Populi itself.

Wednesday January 11, 2006

Filed under: General — m759 @ 11:07 AM
BBC News today:

The image “http://www.log24.com/log/pix06/060111-Reese2.jpg” cannot be displayed, because it contains errors.

Reese Witherspoon
was the winner of
the leading lady award
at the People’s Choice
ceremony in Los Angeles.

“Walk the Line could turn out
to be a monster chick flick,
because its design is
almost mythic….”

Entertainment Weekly     

See the two previous entries.

Related material:
All About Eve.

Wednesday January 11, 2006

Filed under: General — m759 @ 4:09 AM
Chick Flicks
The image “http://www.log24.com/log/pix06/060111-Hen1a.jpg” cannot be displayed, because it contains errors.

From NT Gateway Weblog,
Sunday, Nov. 6, 2005:

 What does Chicken Little
have in common with
The Passion of the Christ?

An anonymous commenter’s answer: “The title character announces the coming of the end, suffers mockery and condemnation, and ends up saving the world through his actions.”

(The “real” answer: “The music for each was composed by John Debney.”)

Related sermon:
Click on the chicken.

Related hymn:

  “Till Armageddon,
no Shalam, no Shalom.
Then the father hen will
  call his chickens home.”

— Johnny Cash

Tuesday, January 10, 2006

Tuesday January 10, 2006

Filed under: General — m759 @ 5:01 AM
Ten is a Hen

From Nov. 12, 2005:

Follow the spiritual journey
that is BEE SEASON.

The image “http://www.log24.com/log/pix05B/051112-Tikkun1.jpg” cannot be displayed, because it contains errors.

“‘Tikkun Olam,
the fixing of the world,’
she whispers.  ‘I’ve been
gathering up the broken vessels
to make things whole again.'”

From Nov. 14, 2005:

Culture Wars

‘Chicken Little’ Lays Golden Egg
(Dean Goodman, Reuters)

‘Bee Season’ Anxiety
(Leonard Klady, Movie City News):

The mixed bag of limited release preems was highlighted by an excellent response to the concert film Sarah Silverman: Jesus is Magic. The film recorded a $19,000 plus per engagement average from seven outings for a $130,000 gross. The family drama Bee Season had a comparable gross but on three times as many screens that translated into anxiety about the Richard Gere film’s expansion prospects.

 Today’s vocabulary lesson:

The image “http://www.log24.com/log/pix06/060110-hendiadys.gif” cannot be displayed, because it contains errors.

Click on the word for the definition.

A search on the related adjective “hendiadic”
leads to an insightful discussion of
religion and law
in contemporary Latin America
by Antônio Flávio Pierucci.

For other material on
Latin America and religion
from Robert Stone and
Nythamar Fernandes de Oliveira,
see the Jan. 25, 2005, entry
Diamonds Are Forever.

Related material:

Yesterday’s link for Nixon’s birthday
 led to an obituary of a Marxist
writer that concluded as follows:

“In 2004, Mr. Magdoff wrote about his friendship with Che Guevara, one of his revolutionary heroes. At what proved to be their final meeting before Mr. Guevara’s death in 1967, Mr. Magdoff asked what he could do to help Cuba. ‘Keep on educating me,’ was the response.”

For the education of Latin America
I recommend the writings of
Pierucci, Stone, and Oliveira,
but not those of Magdoff.

Monday, January 9, 2006

Monday January 9, 2006

Filed under: General — m759 @ 6:01 AM
The image “http://www.log24.com/log/pix06/060109-HappyBday.gif” cannot be displayed, because it contains errors.

Click on picture
for details.

Monday January 9, 2006

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

“In 1782, the Swiss mathematician Leonhard Euler posed a problem whose mathematical content at the time seemed about as much as that of a parlor puzzle. 178 years passed before a complete solution was found; not only did it inspire a wealth of mathematics, it is now a cornerstone of modern design theory.”

— Dean G. Hoffman, Auburn U.,
    July 2001 Rutgers talk

Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):

The image “http://www.log24.com/log/pix06/060109-TarryProof.gif” cannot be displayed, because it contains errors.

Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).

Sunday, January 8, 2006

Sunday January 8, 2006

Filed under: General — m759 @ 8:00 PM

For Stephen Hawking’s Birthday

Epigraphs to the classic novel Cosmic Banditos:

God does not play dice with the universe. –Albert Einstein

Not only does God play dice with the universe, but sometimes he throws them where they cannot be seen. –Stephen Hawking

Today’s Pennsylvania Lottery numbers:

Mid-day 722 7/22, Feast of St. Mary Magdalene.
Evening 399 Page 399, Bartlett’s Familiar Quotations of 1919.

Saturday, January 7, 2006

Saturday January 7, 2006

Filed under: General — m759 @ 11:09 PM

Strange Attractor

The image “http://www.log24.com/log/pix05B/051123-Star.jpg” cannot be displayed, because it contains errors.

Epiphany Star

(See also the star as a
“spider” symbol in the
stories of Fritz Leiber.)

For Heinrich Harrer,
who died today…

The image “http://www.log24.com/log/pix06/060107-WhiteSpider.jpg” cannot be displayed, because it contains errors.

Harrer was one of the 1938 team that first climbed the north face (the Nordwand, also called the Mordwand, or “death” face) of the Eiger.

Wikipedia on the north face of the Eiger:

“A portion of the upper face is called ‘The White Spider,’ as snow-filled cracks radiating from an ice-field resemble the legs of a spider. Harrer used the name for the title of his book about his successful climb, Die Weisse Spinne (translated… as The White Spider).”

Connoisseur of Chaos,”
by Wallace Stevens,
from Parts of a World (1942):


After all the pretty contrast of life and death
Proves that these opposite things partake of one,
At least that was the theory, when bishops’ books
Resolved the world. We cannot go back to that.
The squirming facts exceed the squamous mind,
If one may say so . And yet relation appears,
A small relation expanding like the shade
Of a cloud on sand, a shape on the side of a hill.


The pensive man . . . He sees that eagle float
For which the intricate Alps are a single nest.

Related material:

Friday, January 6, 2006

Friday January 6, 2006

Filed under: General — Tags: — m759 @ 8:23 AM
Today's birthday:
E. L. Doctorow, author of
City of God

"In the garden of Adding,
Live Even and Odd"
City of God

The image “http://www.log24.com/log/pix05B/051202-Cross.gif” cannot be displayed, because it contains errors.
Adapted from
Ad Reinhardt

"… I don't write exclusively on Jewish themes or about Jewish characters. My collection of short stories, Strange Attractors, contained nine pieces, five of which were, to some degree, Jewish, and this ratio has provided me with a precise mathematical answer (for me, still the best kind of answer) to the question of whether I am a Jewish writer. I am five-ninths a Jewish writer."
Rebecca Goldstein,
"Against Logic"
For related remarks,
click on the cross.

Friday January 6, 2006

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


The image “http://www.log24.com/log/pix05B/051123-Star.jpg” cannot be displayed, because it contains errors.

“A related epiphanic question, second only in interest to the question of the nature of epiphany, is how Joyce came by the term. The religious implications would have been obvious to Joyce: no Irish Catholic child could fail to hear of and to understand the name of the liturgical feast celebrated on January 6. But why does Joyce appropriate the term for his literary theory? Oliver St. John Gogarty (the prototype of the Buck Mulligan of Ulysses)… has this to say: ‘Probably Father Darlington had taught him, as an aside in his Latin class– for Joyce knew no Greek– that ‘Epiphany’ meant ‘a shining forth.'”

— William T. Noon, Society of Jesus,
Chapter 4 of Joyce and Aquinas,
Yale University Press, 1957

Epigraphs to The Shining, by Stephen King:

The image “http://www.log24.com/log/pix05B/051231-Shining.gif” cannot be displayed, because it contains errors.

For more about shining, click on the star.

Thursday, January 5, 2006

Thursday January 5, 2006

Filed under: General — m759 @ 6:15 PM
Whirligig (continued)
The image “http://www.log24.com/log06/saved/060105-Kalachakra2.jpg” cannot be displayed, because it contains errors.

“Thus the whirligig of time
brings in his revenges.”
Twelfth Night,
Act V, Sc. I  [text]

See also January 5 in 2003 and 2005.

Thursday January 5, 2006

Filed under: General — m759 @ 9:00 AM
Hamilton’s Whirligig

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

For details, see Visualizing GL(2,p).

“Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….”

— Paul Preuss, Broken Symmetries

Thursday January 5, 2006

Filed under: General — m759 @ 12:00 AM
Dark City

The image “http://www.log24.com/log/pix06/060105-ShellBeach.jpg” cannot be displayed, because it contains errors.

I stood in the cold on the porch
And could not think of anything so perfect
As man’s hope of light in the face of darkness.

— Richard Eberhart,
“The Eclipse”

Wednesday, January 4, 2006

Wednesday January 4, 2006

Filed under: General,Geometry — m759 @ 4:04 AM
Dragon School

In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School.  Carpenter died a year ago today.

From Log24 on Nov. 16, 2005:


Adam Gopnik on C. S. Lewis in the New Yorker:

“Lewis began with a number of haunted images….”

“The best of the books are the ones… where the allegory is at a minimum and the images just flow.”

“‘Everything began with images,’ Lewis wrote….”

The image “http://www.log24.com/log/pix05B/051116-Time.jpg” cannot be displayed, because it contains errors.

From Paul Preuss,
Broken Symmetries
(see previous entry):

“Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….”

Verbum Sat Sapienti?

The image “http://www.log24.com/theory/images/EscherVerbum2.jpg” cannot be displayed, because it contains errors.

Escher’s Verbum

The image “http://www.log24.com/theory/images/DTinvar246.jpg” cannot be displayed, because it contains errors.

Solomon’s Cube

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

Geometry of the I Ching

Wednesday January 4, 2006

Filed under: General — m759 @ 2:00 AM
The Shining

The Shining according to
the Catholic Church:

“The Transfiguration of Christ is the culminating point of His public life…. Jesus took with him Peter and James and John and led them to a high mountain apart, where He was transfigured before their ravished eyes.  St. Matthew and St. Mark express this phenomenon by the word metemorphothe, which the Vulgate renders transfiguratus est.   The Synoptics explain the true meaning of the word by adding ‘his face did shine as the sun: and his garments became white as snow,’ according to the Vulgate, or ‘as light,’  according to the Greek text.  This dazzling brightness which emanated from His whole Body was produced by an interior shining of His Divinity.”

— The Catholic Encyclopedia of 1912

The Shining according to
Paul Preuss:

From Broken Symmetries, 1983, Chapter 16:

“He’d toyed with ‘psi’ himself…. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand– for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox….   

Peter [Slater] had never thirsted after ‘hidden variables’ to explain what could not be pictured.  Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….

Those so-called crazy psychics were too sane, that was their problem– they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry.”

Tuesday, January 3, 2006

Tuesday January 3, 2006

Filed under: General — m759 @ 8:06 PM
Step Three
of January, 2006

From St. Andrew’s Day,

See also Step One —
  “Happy Six,”*
  and Step Two —
  “Then a Miracle Occurs.”**

The miracle occurred on the
Feast of the Transfiguration,
August 6, 1945, at 8:15 AM
in Hiroshima, Japan.

* This Jan. 1 entry links to a paper by Robert T. Curtis  from The Arabian Journal for Science and Engineering (King Fahd University, Dhahran, Saudi Arabia), Volume 27, Number 1A, January 2002.

** This Jan. 2 entry discusses Einstein as, according to the New York Times, “a moral and even spiritual sage.”

Monday, January 2, 2006

Monday January 2, 2006

Filed under: General — m759 @ 12:25 PM

Then a Miracle Occurs

  The New York Times on Sunday,
  New Year’s Day, 2006,
  by John Horgan–
 “Einstein Has Left the Building“–

“Down the hall from my office, Albert Einstein’s electric-haired visage beams from a poster for the ‘World Year of Physics 2005.’ The poster celebrates the centennial of the ‘miraculous year’ when a young patent clerk in Bern, Switzerland, revolutionized physics with five papers on relativity, quantum mechanics and thermodynamics. ‘Help make 2005 another Miraculous Year!’ the poster exclaims…. As 2005 wound down with no miracles in sight, the poster took on an increasingly poignant cast, like a Kerry/Edwards bumper sticker….”

From a debate:

KERRY: “I’m going to be a president who believes in science.”

KERRY: “I’m a Catholic – raised a Catholic. I was an altar boy. Religion has been a huge part of my life, helped lead me through a war, leads me today.”

BUSH: “Trying to decipher that.”

Horgan quotes Einstein:

The image “http://www.log24.com/log/pix06/060102-Einstein.gif” cannot be displayed, because it contains errors.

Click on “religion” in this quote to find out what Einstein really meant.

Here’s a bumper sticker for Horgan:

The image “http://www.log24.com/log/pix06/060102-Sticker.gif” cannot be displayed, because it contains errors.

More from Horgan’s New Year’s Day sermon:

“We revere [Einstein] not only as a scientific genius but also as a moral and even spiritual sage….”

“What you mean ‘we,’ blue man?”
— The Red States

Sunday, January 1, 2006

Sunday January 1, 2006

Filed under: General — m759 @ 6:00 AM

The image “http://www.log24.com/log/pix06/060101-SixOfOne.jpg” cannot be displayed, because it contains errors.

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