Saturday, February 27, 2010

Cubist Geometries

Filed under: Uncategorized — Tags: , — m759 @ 2:01 PM

"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 3x3x3 geometer's cube, with coordinates

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

The 13 symmetry axes of the cube

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

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

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

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

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)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by 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.


Filed under: Uncategorized — m759 @ 10:31 AM
              Tell me about yourself, Linda.
              What do you want to know?
              Anything. I'd just like to know about you.
              Well, basically...
              ...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.

Friday, February 26, 2010

Household Name

Filed under: Uncategorized — m759 @ 9:00 PM

Detail from yesterday's post

Surrealistic Alarm Clock

'Pussy-Footer' alarm clock, LIFE Magazine, Oct. 10, 1949, page 122


Crawdaddy article on 'Surrealistic Pillow,' the classic 1967 album by Jefferson Airplane

"…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.

Ninefold square with shades of gray in chessboard pattern

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

Thursday, February 25, 2010

Through the Blackboard

Filed under: Uncategorized — m759 @ 12:07 PM

Or: "Gopnik Meets Oppenheimer in Heaven"

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

Midrash on "A Serious Man"
by Steven Menashi at
The American Scene

"A Serious Man kicks off with a Yiddish-language frame story that takes place in a 19th-century Eastern European shtetl, where a married couple has an enigmatic encounter with an old acquaintance who may be a dybbuk," recounts Dana Stevens . "The import of this parable is cryptic to the point of inscrutability."

It seems to me that the Coen Brothers’ dybbuk is the Jewish folkloric equivalent of Schrodinger’s Cat .

When we first meet the main character, a physics professor named Larry Gopnik, he’s writing equations on the board: "So if that’s that, then we can do this, right? Is that right? Isn’t that right? And that’s Schrodinger’s paradox, right? Is the cat dead or is the cat not dead?" Likewise, we can’t know whether Fyvush Finkel [the aforementioned old acquaintance] is alive or a dybbuk. We can only evaluate probabilities. When a Korean student named Clive Park complains to Larry that he shouldn’t have failed the Physics midterm because "I understand the physics. I understand the dead cat," Larry says:

You can’t really understand the physics without understanding the math. The math tells how it really works. That’s the real thing; the stories I give you in class are just illustrative; they’re like, fables, say, to help give you a picture. An imperfect model. I mean— even I don’t understand the dead cat. The math is how it really works.

But the fable actually tells us that the math doesn’t capture reality.

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"–

Physicist accelerated against his blackboard in 'A Serious Man'


Blackboard at the Institute for Advanced Study–

J. Robert Oppenheimer at his blackboard

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

(Click to enlarge.)

Oppenheimer homecoming, with ad for 'Pussy-Footer' alarm clock


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.

Wednesday, February 24, 2010


Filed under: Uncategorized — m759 @ 4:24 PM

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"

Tuesday, February 23, 2010

Fish Story

Filed under: Uncategorized — m759 @ 10:12 AM

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

“Like all dreamers I confuse
disenchantment with truth.”
– Jean-Paul Sartre


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).

Monday, February 22, 2010

Annals of Philosophy

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

The Medium is the Message

Marshall McLuhan

From the Wikipedia article
on Marshall McLuhan–

McLuhan 'tetrad' figure with four diamonds surrounding a fifth, the medium

From yesterday

(Click images for some background.)

Ian McKellen at 'Neverwas' diamond windows

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

Sunday, February 21, 2010

Reflections, continued

Filed under: Uncategorized — Tags: — m759 @ 2:02 PM

"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.


Filed under: Uncategorized — m759 @ 12:06 PM

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

Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form ri2 or (rirj)k), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups.

Finite fields

This section requires expansion.

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

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

Journal   Geometriae Dedicata
Publisher   Springer Netherlands
Issue   Volume 9, Number 2 / June, 1980

Ascher Wagner's 1977 dismissal of reflection groups over fields of characteristic 2

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

Clearly the eightfold cube is a counterexample.

Saturday, February 20, 2010

The Mathieu Relativity Problem

Filed under: Uncategorized — m759 @ 10:10 AM

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 in 1931 on his 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

Friday, February 19, 2010

Mimzy vs. Mimsy

Filed under: Uncategorized — m759 @ 11:00 AM

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–

The Eightfold Cube and a related page from a 1906 edition of 'Paradise of Childhood'

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.)

Requiem for a Wizard*

Filed under: Uncategorized — m759 @ 7:20 AM

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

Images from March 3, 2004:

The Jewel
in Venn's Lotus


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).

Thursday, February 18, 2010

Theories: An Outline

Filed under: Uncategorized — Tags: , — m759 @ 10:31 AM

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.)

Tuesday, February 16, 2010

Mysteries of Faith

Filed under: Uncategorized — m759 @ 9:00 AM

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–

2x2x2 cube

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

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

For a more Protestant meditation,
see The Cross of Descartes


Descartes's Cross

"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–

Quine and Derrida at Notre Dame Philosophical Reviews

Notre Dame Philosophical Reviews

Sunday, February 14, 2010

Sunday School

Filed under: Uncategorized — m759 @ 9:00 AM

"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


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.)


Filed under: Uncategorized — m759 @ 8:28 AM

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!"

Saturday, February 13, 2010

Entertainment continued

Filed under: Uncategorized — m759 @ 9:28 AM

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

Professor Arielle Saiber with chess set

Excerpt from Jasper Hopkins's 'Concise Introduction to the Philosophy of Nicholas of Cusa

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)–

Ian J. Thompson, 7 Dec. 2009

Quantum mechanics describes the probabilities of actual outcomes in terms of a wave function, or at least of a quantum state of amplitudes that varies with time. The public always asks what the wave function is, or what the amplitudes are amplitudes of. Usually, we reply that the amplitudes are ‘probability amplitudes’, or that the wave function is a ‘probability wave function’, but neither answer is ontologically satisfying since probabilities are numbers, not stuff. We have already rehearsed the objections to the natural world being made out of numbers, as these are pure forms. In fact, ‘waves’, ‘amplitudes’ and ‘probabilities’ are all forms, and none of them can be substances. So, what are quantum objects made of: what stuff?

According to Heisenberg [6], the quantum probability waves are “a quantitative formulation of the concept of ‘dynamis’, possibility, or in the later Latin version, ‘potentia’, in Aristotle’s philosophy. The concept of events not determined in a peremptory manner, but that the possibility or ‘tendency’ for an event to take place has a kind of reality—a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of the idea or the image—this concept plays a decisive role in Aristotle’s philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature.” Unfortunately Heisenberg does not develop this interpretation much beyond the sort of generality of the above statements, and the concept of ‘potentiality’ remains awkwardly isolated from much of his other thought on this subject [7]. It is unclear even what he means by ‘potentia’.


Heisenberg, W. 1961 On Modern Physics, London: Orion Press.


[6] W. Heisenberg, ‘Planck’s discovery and the philosophical problems of atomic physics’, pp. 3-20 in Heisenberg (1961).

[7] Heisenberg, for example, brings into his thought on quantum physics the Kantian phenomena/noumena distinction, as well as some of Bohr’s ideas on ‘complementarity’ in experimental arrangements.

Friday, February 12, 2010

Capital E

Filed under: Uncategorized — m759 @ 10:30 AM

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….

Venn diagram of three sets

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

Hilary Swank in 'Million Dollar Baby'

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.

Wednesday, February 10, 2010

Mathematics and Religion, continued

Filed under: Uncategorized — m759 @ 12:00 PM

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"–


Sunday, February 7, 2010

Today’s Sermon

Filed under: Uncategorized — m759 @ 11:00 AM

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.


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

Russell discovers his paradox

Click for some context.

Saturday, February 6, 2010

Conceptual Art, continued–

Filed under: Uncategorized — m759 @ 11:01 AM

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

Rebecca Goldstein and a Cullinane quaternion

The added Quaternion  picture is from
Groundhog Day, 2009.

Conceptual Art

Filed under: Uncategorized — m759 @ 9:00 AM

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)

Stephen Savage illustration for 2/2/03 NYT review of 'A Box of Matches'

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

Friday, February 5, 2010

The Great Brown

Filed under: Uncategorized — m759 @ 9:00 PM

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.

Thursday, February 4, 2010

Requiem for a Force–

Filed under: Uncategorized — Tags: — m759 @ 3:30 PM

Where Three Worlds Meet

Venn diagram of three sets

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."


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.

Phenomenology of 256

Filed under: Uncategorized — Tags: — m759 @ 11:30 AM

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:

Quotes from Brautigan's 'The Hawkline Monster' and Hoban's 'Riddley Walker'

Happy birthday, Russell Hoban.

Wednesday, February 3, 2010

Attitude Adjustment

Filed under: Uncategorized — Tags: — m759 @ 1:06 PM

"A generation lost in space"
— American Pie

Sperry F3 attitude gyroscope

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 —

VisualCommander quaternion display from Princeton Satellite Systems

Holly Day

Filed under: Uncategorized — Tags: — m759 @ 6:00 AM

Today's word:

The musical notation 'fermata,' or 'birdseye'


"February made me shiver…."
American Pie

Tuesday, February 2, 2010

Time After Time

Filed under: Uncategorized — m759 @ 12:00 AM

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.

Monday, February 1, 2010

Frame by Frame

Filed under: Uncategorized — m759 @ 7:26 PM

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

“… even if we can break down time into component Walsh functions, what would it achieve?”

– The Professor, in “Passing in Silence,” by Oliver Humpage

“Being is not a steady state but an occulting one: we are all of us a succession of stillness blurring into motion on the wheel of action, and it is in those spaces of black between the pictures that we find the heart of mystery in which we are never allowed to rest. The flickering of a film interrupts the intolerable continuity of apparent world; subliminally it gives us those in-between spaces of black that we crave.”

Gösta Kraken, Perception Perceived: an Unfinished Memoir (p. 9 in Fremder, a novel by Russell Hoban)

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."

For St. Bridget’s Day

Filed under: Uncategorized — m759 @ 12:25 PM

"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.

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