Log24

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Winners: Real and Imaginary

From the archives of the Boot:

(Click to enlarge)

See this morning's post and
this afternoon's New York Times
headline "The Academy Smiles
With Both Faces" (5:26 PM).

See also 5/26.

The Magic Lyre

(Click image for context.)

See also Saturday's post–

as well as Solemn Dance
   and Mazur at Delphi.

(This last is apparently based on
a talk given by Barry Mazur at Delphi
in 2007 and may or may not appear in
a book, Mathematics and Narrative,
to be published in 2010.)

Suggested tune for the lyre–

"Send me the pillow
  that you dream on,"

in memory of Hank Locklin,
who died on this date last year.
 

Meanwhile, back at the AMS–

Ready for His Closeup

Closeup–

For instance…

"You're talking about…
the moment of creation!"

See also
Rebecca Goldstein
and the Number 36

The Magic Lyre

The front page of tomorrow's New York Times Book Review is devoted to a new novel titled Angelology.

Detail of the front page, top right corner–

"…this will be popular for fans of such historical thrillers as… Katherine Neville's The Eight." –Library Journal

The New York Times review is more flattering– "a terrifically clever thriller– more Eco than Brown…."

Related historical remarks–

the symbology novels of Dan Brown and…

TIME magazine cover, issue
dated March 15, 2010–
"History Maker: How Tom Hanks is
redefining America's past"

For some theological background to
this post and today's noon post,
see the use of the word "harrowing"
in this journal — particularly on
April 19, 2003– Holy Saturday.

Alyssa is  Wonderland

Manohla Dargis in The New York Times  yesterday–

"Of course the character of Carroll’s original Alice is evident in each outrageous creation she dreams up in 'Wonderland' and in the sequel, 'Through the Looking-Glass,' which means that she’s a straight man to her own imagination. (She is  Wonderland.)"

From Inside the White Cube–

"The sacramental nature of the space becomes clear, and so does one of the great projective laws of modernism: as modernism gets older, context becomes content. In a peculiar reversal, the object introduced into the gallery 'frames' the gallery and its laws."

From Yogi Berra–

"When you come to a fork in the road, take it."

Related material:  For Baron Samedi and…

Symbology

"And there we were all in one place,
A generation lost in space…"
– Don McLean, "American Pie" 

Today's NY Times says Robert T. McCall, space artist, died at 90 on Feb. 26.

"His most famous image may be the gargantuan mural, showing events from the creation of the universe to men walking on the Moon, on the south lobby wall of the National Air and Space Museum on the National Mall in Washington. More than 10 million people a year pass it.

Or it might be his painting showing a space vehicle darting from the bay of a wheel-shaped space station, which was used in a poster for Stanley Kubrick’s landmark 1968 film, '2001: A Space Odyssey.'"

Cover art by McCall, with autograph dated
8/19/05, from a personal web page

Hal in "2010"– "Will I dream?"

Log24 on the day that McCall died–

"Which Dreamed It?"
– Title of final chapter,Through the Looking Glass

"Go ask Alice… I think she'll know."
– Grace Slick, 1967  

Related material: James Joyce in this journal–

"Space: what you damn well have to see."

""Against stupidity the gods themselves contend in vain."
("Mit der Dummheit kämpfen Götter selbst vergebens.")
– Friedrich Schiller (1759-1805) German poet, playwright, critic,
Die Jungfrau von Orleans [The Maid of Orleans], Act III, sc. vi (1801)

To be fair and balanced, we also offer a contrary view–

"Weird Science—Why Editors Must Dare to be Dumb,"
by K.C. Cole, Columbia Journalism Review, 2006

Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008–

"Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…."

Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and yesterday's post–

This is the group of "8 rigid motions
generated by reflections in midplanes"
of Solomon's Cube.

Related material:

"… the action of G₁₆₈ in its alternative guise as SL(3; Z=2Z) is also now apparent. This version of G₁₆₈ was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker."

– Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998

Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber.

*H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1896. Reprinted by Chelsea, New York, 1961.

From the conclusion of Weyl's Symmetry –

One example of Weyl's "structure-endowed entity" is a partition of a six-element set into three disjoint two-element sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.

The automorphism group of this faces-partition contains an order-8 subgroup that is isomorphic to the abstract group C₂×C₂×C₂ of order eight–

The action of Klein's simple group of order 168 on the Cayley diagram of C₂×C₂×C₂ in yesterday's post furnishes an example of Weyl's statement that

"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"

The current article on group theory at Wikipedia has a Rubik's Cube as its logo– 

The article quotes Nathan C. Carter on the question "What is symmetry?"

This naturally suggests the question "Who is Nathan C. Carter?"

A search for the answer yields the following set of images…

Click image for some historical background.

Carter turns out to be a mathematics professor at Bentley University.  His logo– an eightfold-cube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.
 

"The cube has…13 axes of symmetry:
  6 C₂ (axes joining midpoints of opposite edges),
4 C₃ (space diagonals), and
3C₄ (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

A later version of the 13-point plane
by Ed Pegg Jr.–

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

              Tell me about yourself, Linda.
              What do you want to know?
              Anything. I'd just like to know about you.
              Well, basically...
              Yes?
              ...I'm an actress.
              That's wonderful.
              I like drama. I study.
              Yes? Where's that?
              Paul DeLucca. Have you ever heard of him?
              Paul DeLucca? No, but then I wouldn't.
              He's really well-known. He's a genius.
              I'm sure.
              He says he thinks I'm going to make it big.
              I know you will.
              Maybe you've seen some of my movies.
              It's possible.
              Did you ever see The Enchanted Pussy?
              Not yet. But I... I... it's on my list.
              They're videotaped, so you could rent it.

              -- Mighty Aphrodite

                   See also Fish Story.

Detail from yesterday's post–

Surrealistic Alarm Clock

_________________________________________________________

THE SEQUEL:

"…and Surrealistic Pillow  became
a household name
in the house of rock ‘n’ roll."

– Denise Sullivan in Crawdaddy,
October 8, 2009

Related material:

"Which Dreamed It?"
– Title of final chapter,
Through the Looking Glass

"Go ask Alice…
I think she'll know."
– Grace Slick, 1967   

The Crawdaddy  date Oct. 8, 2009
leads to the Log24 post
Graphic Austerity.

Clicking on those words
  in that post will lead you to…
The Logic of Dreams.

Or: "Gopnik Meets Oppenheimer in Heaven"

(Or, for those less philosophically minded, "Raiders of the Lost Pussy")

The story in images below summarizes a meditation suggested by this parable and by

1. Tuesday's post "Fish Story"
    
2. Today's AP thought:
   "Open-mindedness is not the same as empty-mindedness." –John Dewey
    
3. "Zen mind, empty mind."
    
4. Today's NY Times obituary for Selma G. Hirsh,
   author of The Fears Men Live By (Harper, 1955).
   Hirsh died on St. Bridget's Day.
    
5. A search for the Hirsh book that led to a web page
   with a 1955 review of J. Robert Oppenheimer's book The Open Mind
    
6. A search for the Oppenheimer book that led to
   LIFE magazine's issue of Oct. 10, 1949
    
7. "Satori means 'awakening.'" — TIME magazine, Nov. 21, 1960

Blackboard in "A Serious Man"–

Blackboard at the Institute for Advanced Study–

"Daddy's home! Daddy's home!"

(Click to enlarge.)

Related material–

A Zen meditation from Robert Pirsig
is suggested by the time on the above
alarm clock– 8:20– interpreted,
surrealistically, as a date — 8/20.

A topic related to A Simple Reflection Group of Order 168–

Transvection groups over GF(2). See, for instance…

1. Binary Coordinate Systems, by Steven H. Cullinane, 1984
    
2. Classification of the Finite N-Generator Transvection Groups Over Z₂, by Jizhu Nan and Jing Zhao, 2009, Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010), 185–202
    
3. Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"

Stanley Fish reviewed a new book, Steven Smith's The Disenchantment of Secular Discourse, in yesterday's online NY Times–

…the self-impoverished discourse of secular reason does in fact produce judgments, formulate and defend agendas, and speak in a normative vocabulary. How is this managed? By “smuggling,” Smith answers.

. . . the secular vocabulary within which public discourse is constrained today is insufficient to convey our full set of normative convictions and commitments. We manage to debate normative matters anyway— but only by smuggling in notions that are formally inadmissible, and hence that cannot be openly acknowledged or adverted to.

The notions we must smuggle in, according to Smith, include “notions about a purposive cosmos, or a teleological nature stocked with Aristotelian ‘final causes’ or a providential design,” all banished from secular discourse because they stipulate truth and value in advance rather than waiting for them to be revealed by the outcomes of rational calculation. But if secular discourse needs notions like these to have a direction— to even get started— “we have little choice except to smuggle [them] into the conversations— to introduce them incognito under some sort of secular disguise.”

And how do we do that?

A Jewish Answer

By the Coen brothers in "A Serious Man"–

"When the truth is found to be lies
And all the joy within you dies…."

A Christian answer

Disenchantment author Steven Smith is a a professor at the University of San Diego. This suggests a look at the feast day of San Diego himself… Here are Log24 posts that mention that day, November 12 (which is also Grace Kelly's birthday).

The Medium is the Message

Marshall McLuhan

From the Wikipedia article
on Marshall McLuhan–

From yesterday–

(Click images for some background.)

Related material:

Feast of St. Louis, 2003,

a web page on McLuhan's
student Walter J. Ong, S. J.,

and Jung and the Imago Dei

"The eye you see him with is the same
eye with which he sees you."

– Father Egan on page 333
of Robert Stone's A Flag for Sunrise
(Knopf hardcover, 1981)

Part I– Bounded in a Nutshell

Ian McKellen at a mental hospital's diamond-shaped window in "Neverwas"

Part II– The Royal Castle

Ian McKellen at his royal castle's diamond-shaped window in "Neverwas"

Part III– King of Infinite Space

H.S.M. Coxeter crowns himself "King of Infinite Space"

Related material:

See Coxeter in this journal.

From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005– as revised on Nov. 25, 2009–

Related material:

"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and

"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"

by Ascher Wagner, U. of Birmingham, received 27 July 1977

[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]

Clearly the eightfold cube is a counterexample.

Weyl on what he calls the relativity problem–

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

– Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

– Hermann Weyl, 1946, The Classical Groups, Princeton University Press, p. 16

Twenty-four years ago a note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M₂₄ (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M₂₄.  That is, there is no obvious way to apply exactly 24 distinct transformable coordinates (or symbol-strings) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M₂₄.

There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M₂₄.

R.D. Carmichael on his 1931 construction of the Steiner system S(5,8,24)–

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

– R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Deep Play:
Mimzy vs. Mimsy

From a 2007 film, "The Last Mimzy," based on
the classic 1943 story by Lewis Padgett
  "Mimsy Were the Borogoves"–

As the above mandala pictures show,
the film incorporates many New Age fashions.

The original story does not.

A more realistic version of the story
might replace the mandalas with
the following illustrations–

Click to enlarge.

For a commentary, see "Non-Euclidean Blocks."

(Here "non-Euclidean" means simply
other than  Euclidean. It does not imply any
  violation of Euclid's parallel postulate.)

In memory of
Susanna Kaysen's father,
who died on February 8–

Images from March 3, 2004:

The Jewel
in Venn's Lotus

and

El Pato-lógico and a

Dream of Heaven

* The title of the 2004 post containing these images is "Deep Play." For some notion of the depth of the play "The Life of Carl Kaysen," see "The Kaysen Memos," pp. 271-278 in James Carroll's House of War  (1st ed. May 16, 2006).

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

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

– Coxeter, 1987, introduction to Trudeau's The Non-Euclidean Revolution

1.1 Trudeau's Diamond Theory of Truth
1.2 Trudeau's Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–
          Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
          Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–
           Examples: Eightfold Cube and Solomon's Cube
2.2.2 The Algebra–
           Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+"characteristic two" (or "2") and for generators+relations+"GF(2)". (This last search is the source of the Grams reference in 2.2.2 above.)
 

From today's NY Times–

Obituaries for mystery authors
Ralph McInerny and Dick Francis

From the date (Jan. 29) of McInerny's death–

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"

– Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson

From the date (Feb. 14) of Francis's death–

The EIghtfold Cube

The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.

This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.

For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
  and the death of Coxeter–

Putting Descartes Before Dehors

For a more Protestant meditation,
see The Cross of Descartes–

"I've been the front end of a horse
and the rear end. The front end is better."
– Old vaudeville joke

For further details, click on
the image below–

Notre Dame Philosophical Reviews

"Simplify, simplify." — Henry David Thoreau

"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."

– Alexandre Borovik, 2010 (See previous entry.)

Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page

A Simple Reflection Group
of Order 168.

Background:

Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups–

15.1 (p. 114): Every finite reflection group is a Coxeter group.

15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.

Consider in this context the above simple reflection group of order 168.

(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" –A.M. Cohen.)

From Alexandre Borovik's new book
Mathematics Under the Microscope
  (American Mathematical Society, 2010)–

Related material:

Finite Geometry and Physical Space
(Good Friday, 2009)

This kindergarten-level discussion of
the simple group of order 168
also illustrates Thoreau's advice:
"Simplicity, simplicity, simplicity!"

“Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning,
   Harvard University Press, 2004

Geometry of Language,
continued from St. George's Day, 2009–

Related material:

Prima Materia,
The Galois Quaternion,
and The Wake of Imagination.

See also the following from a physicist
(not of the most orthodox sort, but his remarks
  here on Heisenberg seem quite respectable)–

Where Entertainment is God, continued–

The following paragraphs are from a review by Piotr Siemion of Infinite Jest, a novel by David Foster Wallace. Illustrations have been added.

"Wallace was somehow able to twist together three yarns…. …there's a J.D Salinger for those who like J.D. Salinger. There's William Burroughs for those hardy souls who like some kick in their prose. And there's a dash of Kurt Vonnegut too. All three voices, though, are amplified in Infinite Jest beyond mere distortion and then projected onto Wallace's peculiar own three-ring circus….

… there's entertainment. Make it a capital E.

Illustration by Clint Eastwood
from Log24 post "E is for Everlast"

Infinite Jest revolves, among its many gyrations, around the story of the Entertainment, a film-like creation going by the title of 'Infinite Jest' and created shortly before his suicidal death by the young tennis star's father. The Entertainment's copies are now being disseminated clandestinely all over Wallace's funny America. Problem is, of course, that the film is too good. Anybody who gets to watch it becomes hooked instantly and craves only to watch it again, and again, and again, until the audience drops dead of exhaustion and hunger. Why eat when you're entertained by such a good movie? Wallace's premise brings you back to that apocryphal lab experiment in which rats were treated to a similar choice. When the rat pushed one button, marked FOOD, it would get a food pellet. The other button, marked FUN, would fire up an electrode rigged right into the orgasm center somewhere in the rat's cortex. Needless to add, one rat after another would drop dead from hunger, still twitching luridly and trying to finesse one last push of the button. Same thing in Wallace's story, especially that even those characters who have not seen the Entertainment yet, keep on entertaining themselves by different means."

The title of the Entertainment, "Infinite Jest," might also be applied to a BBC program featuring mathematician Peter J. Cameron. The program's actual title was "To Infinity and Beyond." It was broadcast the night of Feb. 10 (the date of this journal's previous post).

Few, however, are likely to find the Infinity program addictive. For closer approaches to Wallace's ideal Entertainment, see instead Dante (in the context of this journal's Feb. 4 posts on Cameron and the afterlife) and the BBC News.

But Seriously…

From "Georg Cantor and the Battle for Transfinite Set Theory," by Joseph W. Dauben (pdf)–

"It is easy, of course, to misinterpret the religious element in Cantor's thinking, as popularizers often do. This was certainly the case in an article that appeared not long ago in the French magazine La Recherche, which supplied [the above] caricatures to illustrate an expository article about Cantor, his religious convictions, psychological illness and transfinite set theory.* The first drawing depicts Cantor in ecstasy, as it were, receiving the divine message. In the second illustration, the figure with the gun of course is meant to be Kronecker– with God helping Cantor to maintain his balance– all of which rests precariously on a transfinite aleph. But there is a very serious side to all of this…."

* Pierre Thuillier, “Dieu, Cantor et l'Infini,” La Recherche, (December, 1977), pp. 1110-1116.

Everything and More: A Compact History of Infinity, by David Foster Wallace–

"In modern medical terms, it's fairly clear that G.F.L.P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. Of course, this makes for less interesting flap copy than Genius Driven Mad by Attempts to Grapple with ∞. The truth, though, is that Cantor's work and its context are so totally interesting and beautiful that there's no need for breathless Prometheusizing of the poor guy's life. The real irony is that the view of ∞ as some forbidden zone or road to insanity– which view was very old and powerful and haunted math for 2000+ years– is precisely what Cantor's own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George's loss to the dragon: it's not only wrong but insulting."

Related entertainment:

David Foster Wallace,
Influential Writer, Dies at 46

and the film "Neverwas"–

Mathematics and Religion, continued–

Calvin Jongsma, review of an anthology titled Mathematics and the Divine–

"Believers of many faiths have found significant points of contact between their religious outlooks and mathematics. Not all of these claims were made in the distant past or by certified crackpots…."

Edward Nelson in "Warning Signs of a Possible Collapse of Contemporary Mathematics"–

"The most impressive feature of Cantor’s theory is that he showed that there are different sizes of infinity, by his famous diagonal argument. But Russell applied this argument to establish his paradox: the set of all sets that are not elements of themselves both is and is not an element of itself."

Jongsma's assertion appears to be true. Nelson's appears to be false. Discuss.

Remarks:

Saying that someone applied some argument– any argument will do here– to establish a paradox– any paradox will do here– casts into doubt the validity of either the argument, the application of the argument, or both. In the Cantor-Russell case, such doubt is unnecessary, since the paradox is clearly independent of the diagonal argument. There is certainly an historical connection between Cantor's argument and Russell's paradox– see, for instance, Wikipedia on the latter. The historical connection is, however, not a logical connection.

For Russell discovering his paradox without the use of Cantor's diagonal argument, see Logicomix–

Click for some context.

Argument for the Existence of Rebecca

Adapted from YouTube's "Mathematics and Religion," starring Rebecca Newberger Goldstein, author of the recent novel 36 Arguments for the Existence of God–

The added Quaternion  picture is from
Groundhog Day, 2009.

The Plane of Time

From tomorrow's NY Times Book Review, Geoff Dyer's review of DeLillo's new novel Point Omega is now online–

"The book begins and ends with Douglas Gordon’s film project '24 Hour Psycho' (installed at the Museum of Modern Art in Manhattan in 2006), in which the 109-­minute Hitchcock original is slowed so that it takes a full day and night to twitch by. DeLillo conveys with haunting lucidity the uncanny beauty of 'the actor’s eyes in slow transit across his bony sockets,' 'Janet Leigh in the detailed process of not knowing what is about to happen to her.' Of course, DeLillo being DeLillo, it’s the deeper implications of the piece— what it reveals about the nature of film, perception and time— that detain him. As an unidentified spectator, DeLillo is mesmerized by the 'radically altered plane of time': 'The less there was to see, the harder he looked, the more he saw.'

This prologue and epilogue make up a phenomenological essay on one of the rare artworks of recent times to merit the prefix 'conceptual.'"

Related material:

Steering a Space-Plane
(February 2, 2003)

Holly Day
(February 3, 2010)

Attitude Adjustment
(February 3, 2010)

Cover illustration by Stephen Savage,
NY Times Book Review,
Feb. 2 (Candlemas), 2003

“We live the time that a match flickers.”

– Robert Louis Stevenson, Aes Triplex

Today's New York Times on a current theatrical presentation of The Great Gatsby–

"Throughout the show, the relationship between what is read and its context keeps shifting, with the real world finally giving way entirely to the fictive one."

Owl Eyes in The Great Gatsby–

"This fella's a regular Belasco."

David Brown, producer. Brown died on Monday.

From The Diamond as Big as the Monster in this journal on Dec. 21, 2005–

"At the still point, there the dance is.” –T. S. Eliot, Four Quartets

Eliot was 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

Related material: Yesterday's posts and the jewel in Venn's lotus.

Where Three Worlds Meet

From an obituary for David Brown, who died at 93 on Monday–

"David Brown was a force in the entertainment, literary and journalism worlds," Frank A. Bennack, Jr., vice chairman and chief executive officer of Hearst Corporation, said in a statement Tuesday. –Polly Anderson of the Associated Press

Mark Kramer, "Breakable Rules for Literary Journalists," Section 8–

"Readers are likely to care about how a situation came about and what happens next when they are experiencing it with the characters. Successful literary journalists never forget to be entertaining. The graver the writer's intentions, and the more earnest and crucial the message or analysis behind the story, the more readers ought to be kept engaged. Style and structure knit story and idea alluringly.

If the author does all this storytelling and digressing and industrious structure-building adroitly, readers come to feel they are heading somewhere with purpose, that the job of reading has a worthy destination. The sorts of somewheres that literary journalists reach tend to marry eternal meanings and everyday scenes. Richard Preston's 'The Mountains of Pi,' for instance, links the awkward daily lives of two shy Russian emigre mathematicians to their obscure intergalactic search for hints of underlying order in a chaotic universe."

Hints:

“Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning, Harvard U. Press, 2004

"According to the Buddha, scholars speak in sixteen ways of the state of the soul after death…. While I hesitate to disagree with the Compassionate One, I think there are more than sixteen possibilities described here…."

– Peter J. Cameron today

"That's entertainment!"

– Jack Haley Jr.

From Peter J. Cameron's weblog today–

According to the Buddha,

Scholars speak in sixteen ways of the state of the soul after death. They say that it has form or is formless; has and has not form, or neither has nor has not form; it is finite or infinite; or both or neither; it has one mode of consciousness or several; has limited consciousness or infinite; is happy or miserable; or both or neither.

He does go on to say that such speculation is unprofitable; but bear with me for a moment.

With logical constructs such as “has and has not form, or neither has nor has not form”, it is perhaps a little difficult to see what is going on. But, while I hesitate to disagree with the Compassionate One, I think there are more than sixteen possibilities described here: how many?

Cameron's own answer (from problem solutions for his book Combinatorics)–

One could argue here that the numbers of choices should be multiplied, not added; there are 4 choices for form, 4 for finiteness, 2 for modes of consciousness, 2 for finiteness of consciousness, and 4 for happiness, total 2⁸ = 256. (You may wish to consider whether all 256 are really possible.)

Related material– "What is 256 about?"

Some partial answers–

April 2, 2003 — The Question (lottery number)

May 2, 2003 — Zen and Language Games (page number)

August 4, 2003 — Venn's Trinity (power of two)

September 28, 2005 — Mathematical Narrative (page number)

October 26, 2005 — Human Conflict Number Five (chronomancy)

June 23, 2006 — Binary Geometry (power of two)

July 23, 2006 — Partitions (power of two)

October 3, 2006 — Hard Lessons (number of pages,
                                 as counted in one review)

October 10, 2006 — Mate (lottery number)

October 8, 2008 — Serious Numbers (page number)

Quoted here Nov. 10, 2009–

Epigraphs at
Peter Cameron’s home page:

Happy birthday, Russell Hoban.

"A generation lost in space"
– American Pie

Sperry F3 attitude gyroscope

Click image for details.

See also the concepts of inner-direction
and other-direction in The Lonely Crowd
by David Riesman et al.  Riesman was,
according to Harvard Square Library,
a contract termination lawyer for
Sperry Gyroscope before turning
to sociology.

EXERCISE — Discuss inner- and
other-direction in education and
in journalism, using the material
in Monday's entry on the
New York Times dunce cap –

  — contrasted with the webpage
excerpted below –

Today's word:

fermata

"February made me shiver…."
– American Pie

Godmother and Cinderella

Meryl Streep and Amy Adams in "Julie & Julia"

The image is from gossipsauce.com on August 2, 2009.

For a darker Godmother/Cinderella pair,
see the film discussed in this journal
on that same date (Lughnasa 2009).

A thought from Pynchon's Against the Day quoted here on Groundhog Day a year ago today–

“We thus enter the whirlwind. It becomes the very essence of a refashioned life, providing the axes to which everything will be referred. Time no long ‘passes,’ with a linear velocity, but ‘returns,’ with an angular one…. We are returned to ourselves eternally, or, if you like, timelessly.”

“Born again!” exclaimed a Christer in the gathering, as if suddenly enlightened.

From "Time's Breakdown," September 17, 2003–

This flashback was suggested by

1. A review in next Sunday's New York Times Book Review of a new novel, Point Omega, by Don DeLillo. The review's title (for which the reviewer, Geoff Dyer, should not be blamed) is "A Wrinkle in Time." The review and the book are indeed concerned with time, but the only apparent connection to the 1962 novel of Madeleine L'Engle also titled A Wrinkle in Time is rather indirect– via the Walsh functions mentioned above.
2. A phrase in the Times's review, "frame by frame," also appeared in this jounal on Saturday. It formed part of the title of a current exhibition at Harvard's Carpenter Center for the Visual Arts.
3. The Carpenter Center exhibition will have an opening reception on February 4.
4. February 4 is also the birthday of the above Russell Hoban, who will turn 85. See a British web page devoted to that event.

DeLillo is a major novelist, but the work of Hoban seems more relevant to the phrase "frame by frame."

"But wait, there's more!"
– Stanley Fish, NY Times Jan. 28

From the editors at The New York Times who, left to their own devices, would produce yet another generation of leftist morons who don't know the difference between education and entertainment–

A new Times column starts today–

The quality of the column's logo speaks for itself. It pictures a cone with dashed lines indicating height and base radius, but unlabeled except for a large italic x to the right of the cone. This enigmatic variable may indicate the cone's height or slant height– or, possibly, its surface area or volume.

Instead of the column's opening load of crap about numbers and Sesame Street, a discussion of its logo might be helpful.

The cone plays a major role in the historical development of mathematics.

Some background from an online edition of Euclid–

"Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere."

For Archimedes and the surface area of a sphere, see (for instance) a discussion by Kevin Brown. For more material on Archimedes, see "Archimedes: Volume of a Sphere," by Doug Faires (2001)– Archimedes' heuristic argument from mechanics that involves the volume of a cone– and Archimedes' more rigorous approach in The Works of Archimedes, edited by T. L. Heath (1897).

The work of Euclid and Archimedes on volumes was, of course, long before the discovery of calculus.  For a helpful discussion of cone volumes involving high-school-level calculus, see, for instance,  the following–

The Times editors apparently feel that
few of their readers are capable of
such high-school-level sophistication.

For some other geometric illustrations
perhaps more appealing than the Times's

dunce cap, see the symbol of
  today's saint– a Bridget Cross–
and a web page on
visualized quaternions.

From The New York Times –

4^TH NIGHT FREE

From this journal –

Sunday, November 5, 2006

Sybil on the Beach

"I was waiting for you," said the young man. "What's new?"

"What?" said Sybil.

"What's new? What's on the program?"

"Animation tends to be a condensed art form, using metamorphosis and metaphor to collide and expand meaning. In this way it resembles poetry."

– Harvard's Carpenter Center for the Visual Arts,
   description of an exhibition–

FRAME BY FRAME: ANIMATED AT HARVARD

January 28–Feb 14, 2010

For example–

Animation – The Animated Diamond Theorem,
                      now shown frame by frame for selected frames

Poetry–

Part I –  "That Nature is a Heraclitean Fire…."

Part II – Metaphor on the covers of a Salinger book–

Click image for details.

For other thoughts on
metamorphosis and metaphor,
see Endgame.

Part I:

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.' She shows deftly and delicately that the void inside Keats’s urn, Heidegger’s jug, or Wallace Stevens’s jar forms the center around which we tend to organize our worlds."

– Harvard University Press on Persons and Things (April 30, 2008), by Barbara Johnson

Part II:

"Did you see more glass?"

Part III:

From the date of Barbara Johnson's death:

"Mathematical relationships were
enough to satisfy him, mere formal
relationships which existed at
all times, everywhere, at once."

– Broken Symmetries, 1983

The X's refer to the pattern on the
cover of a paperback edition
  of Nine Stories, by J. D. Salinger.
Salinger died on Wednesday.

"You remember that book he sent me
from Germany? You know–
those German poems."

In Germany, Wednesday was
Holocaust Memorial Day, 2010.

Home Delivery

"But wait, there's more!"
– Stanley Fish, NY Times today

For a larger image, click on The Catcher.

"Anyway, I keep picturing all these little kids playing some game in this big field of rye and all.  Thousands of little kids, and nobody's around– nobody big, I mean– except me.  And I'm standing on the edge of some crazy cliff.  What I have to do, I have to catch everybody if they start to go over the cliff– I mean if they're running and they don't look where they're going I have to come out from somewhere and catch them.  That's all I'd do all day.  I'd just be the catcher in the rye and all.  I know it's crazy, but that's the only thing I'd really like to be."

– J.D. Salinger, The Catcher in the Rye, Chapter 22

This journal on Aug. 17, 2008:

That post linked to an earlier post illustrating
the triangle formed by Harvard, by the
Mystic River at Somerville, and
by Bunker Hill Community College–

That post also linked to the Wikipedia article
Triangulation, which now states that
"Some members of the U.S. Democratic
Party, in particular the left, insist that
triangulation is 'dead.'"

Perhaps. Click the image below
for some background.

For a view of Somerville from Harvard for Zinn,
see May 31, 2006. For a view of Summertime
for Auchincloss, see the NY Times obituary
of a political figure who died on Sunday.

On that day, this journal pictured a different
 metaphor from Robert Stone's Father Egan–
the jewel in the lotus–

Euclid's classic construction
of the equilateral triangle
offers a different view of
the jewel in Venn's lotus–

For a more poetic approach to
this metaphor, see Log24 on
another Sunday– July 1, 2007.

Happy birthday, Rick Warren.

Yesterday's post may, if one likes, be regarded
  as a nod to Dionysus, god of tragedy.

Here is a complementary passage:

Related material:
Jung and the Imago Dei

From this journal:

Transition to the
Garden of Forking Paths–

(See For Baron Samedi)–

The Found Symbol

and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–

Pages 354-355
On the mirror-play of the fourfold

Pages 356-357
Shaking up a whole culture

Pages 358-359
Cornerstone and crossroads

Pages 360-361
A deep impression embedded in stone

Pages 362-363
A certain Y, a certain V

Pages 364-365
The world is Zeus's play

Page 366
It was necessary to begin again

Recent Log24 entries on Hamlet suggest a look at Giorgio de Santillana's Hamlet's Mill on the Web.

There is a useful transcription by Clifford Stetner. See also an excellent review of Hamlet's Mill by Cecilia Payne-Gaposchkin.

The work of Giorgio de Santillana (like that of Stetner) suggests in turn the following recently quoted advice–

"…you should read Middlemarch every five years or so. Every time… it's a different book, and an even more powerful one."

– Robert Weisbuch, quoted at Critical Mass on Jan. 23.

Related material: Mr. Casaubon and the Key to All Mythologies.

For a simpler key, see On Linguistic Creation.

Sting as Hamlet

Detail, cover of Synchronicity album, 1983

Further details of the text on the album cover–

More Than Matter

"In one or other of G. F. H. Shadbold's two published notebooks, Beyond Narcissus and Reticences of Thersites, a short entry appears as to the likelihood of Ophelia's enigmatic cry: 'Oh, how the wheel becomes it!' referring to the chorus or burden 'a-down, a-down' in the ballad quoted by her a moment before, the aptness she sees in the refrain."

– First words of Anthony Powell's novel "O, How the Wheel Becomes It!" (See Library Thing.)

Related material:

Photo uploaded on January 14, 2009
with caption "This nothing's more than matter"

and the following nothings from this journal
on the same date– Jan. 14, 2009–

Continued from yesterday
and from March 28, 2008–

Being and Nothingness

Jean Simmons and Laurence Olivier, 1948

"Ophelia’s story becomes the Story of O…."
– Elaine Showalter, "Representing Ophelia:
Women, Madness, and the
Responsibilities of Feminist Criticism."
Pp. 220-238 in
Hamlet. Ed. Susanne L. Wofford.
Boston: Bedford Books, 1994.

Yesterday's Times –

Today's Times –

   Annals of Deconstruction –

Click on image for background.

Related material
   for Baron Samedi –

The Found Symbol

"…something is rotten in Denmark. And nothing I have read in the news headlines is going to solve it.

Rant over.

Back to our regular programming."

– A Circle of Quiet today

Our regular programming:

Anne Applebaum in the current New York Review of Books on Arthur Koestler–

"At the moment, he still seems like yesterday's man, unfashionable and obsolete."

Rather like God. See this journal yesterday– Darkness at Noon.

See also David Levine's portrait of Koestler (Dec. 30, 2009)–

– and an objective correlative to yesterday's post –

Click to enlarge.

Church Emblem

Related material:

Faust in Copenhagen,
Meta Physics,
and yesterday's entry.

A NY Times review dated Jan. 20 has the headline

Trying to Paint the Deity by Numbers
Against a Backdrop of Jewish Culture

By JANET MASLIN

"…this novel’s bracing intellectual energy never flags. Though it is finally more a work of showmanship than scholarship, it affirms Ms. Goldstein’s position as a satirist…."

The title of the book under review is
36 Arguments for the
Existence of God: A Work of Fiction.

Related "by the numbers" material–

From the I Ching, commentaries on the lines of Hexagram 36–

"Here the Lord of Light is in a subordinate place and is wounded by the Lord of Darkness…."

"The dark power at first held so high a place that it could wound all who were on the side of good and of the light. But in the end it perishes of its own darkness, for evil must itself fall at the very moment when it has wholly overcome the good, and thus consumed the energy to which it owed its duration."

The Times review of 36 Arguments notes that the book's chapters of fiction number 36, as do the 36 philosophical arguments in the book's title and appendix.

The reviewer– "So much for structure. It is not Ms. Goldstein’s strong suit…."

Some structure related to the above occurrence of 36 in the I Ching–

Another example of eightfold symmetry:

The Large Hadron Collider

See also Angels & Demons in
Hollywood and in this journal.

"I need a man who knows
how the story goes."
– Shania Twain

"The causal story is that a crippled epistemology leads to fanaticism, which then leads to the urge for governmental control…

It is only through gaining control of a state in the modern era that a fanatical group could expect to exclude contrary views and thereby maintain the crippled epistemology of their followers. With the power of a state behind them, they can coerce."

– P. 18 in Russell Hardin, "The Crippled Epistemology of Extremism" (Pp. 3-22 in Political Rationality and Extremism, Albert Breton et al., eds., Cambridge University Press, 2002).

This is the source of more recent uses of the phrase "crippled epistemology."

Cognitive Infiltration

"Conspiracy theorists are not likely to be persuaded by an attempt to dispel their theories; they may even characterize that very attempt as further proof of the conspiracy. Because those who hold conspiracy theories typically suffer from a crippled epistemology, in accordance with which it is rational to hold such theories, the best response consists in cognitive infiltration of extremist groups."

– Cass R. Sunstein, Harvard Law professor,
abstract of "Conspiracy Theories," Jan. 15, 2008

Critical Mass Monday, Jan. 18, 2010, 7:06  AM–

Here you've got three high-powered academics… all engaged in highly pernicious and mutually enabling forms of professional dishonesty. Sunstein comes up [with] what Timothy Burke calls a "consensus-politics liberal-leaning version of COINTELPRO," — Erin O'Connor

Critical Mass? Campaigns Work to Get Voters to Polls

– Washington Wire:
Political Insight and Analysis from
The Wall Street Journal Capital Bureau
January 19, 2010, 11:17 AM ET

David Weigel, "Attacks on Sunstein Frustrate Conservative Fans,"
The Washington Independent, 9/9/09–

"The campaign against Sunstein has largely written itself."