Log24

(Continued from Deconstructing Alice)

The Dream of the Expanded Field

"Somehow it seems to fill my head
with ideas— only I don't exactly know
what they are!"

See also Deep Play.

A New Yorker  weblog post from yesterday, All Souls' Day—

"As the mathematician Terence Tao has written,
math study has three stages:
the 'pre-rigorous,' in which basic rules are learned,
the theoretical 'rigorous' stage, and, last and most intriguing,
'the post-rigorous,' in which intuition suddenly starts to play a part."

Related material— 

Rigor  in a Log24 post of Sunday evening, May 25, 2008: "Hall of Mirrors."

Note in that post the tesseract  viewed as the lattice of
the 16 subsets of a 4-element set.

Some further material related to tesseracts and time, in three stages
(roughly corresponding to Tao's, but not in chronological order): 

1. Bakhtin, 
2. Spaces as Hypercubes, and 
3. Pindar.

See also a recent Log24 post on remarks from Four Quartets .

(The vertices of a tesseract form, in various natural ways, four quartets.)

For clergymen who embrace Trudeau's
"Story Theory of Truth" (see last evening's
7:20 PM post on geometry and A Wrinkle in Time )…

Here are two meditations suggested by
yesterday evening's New York Lottery :

1.  Page 141 in Philosophies of India—

2.  Post 4658 in this journal— A Wrinkle in Dimensions.

(Continued from previous TARDIS posts)

Summary: A review of some  posts from last August is suggested by the death,
reportedly during the dark hours early on October 30, of artist Lebbeus Woods.

An (initially unauthorized) appearance of his work in the 1995 film
Twelve Monkeys …

 … suggests a review of three posts from last August.

"It was a dark and stormy night."— A Wrinkle in Time

From a Sept. 17, 2010, post of Peter J. Cameron
that was linked to here at 8 AM ET today—

This should read instead…

Related material from this journal in 2003—

For Sergeant-Major America—

The image is from posts of Feb. 20, 2011, and Jan. 27, 2012.

This instance of the omega point is for a sergeant major
who died at 92 on Wednesday, October 10, 2012.

See also posts on that date in this journal—

Midnight,  Ambiguation,  Subtitle for Odin's Day,  and
Melancholia, Depression, Ambiguity.

See Galois Ambiguity in this  journal.

The subtitle of Jack Kerouac's novel Doctor Sax
is Faust Part Three.

Related material—

Types of Ambiguity— Galois Meets Doctor Faustus
(this journal, December 14, 2010).

See also tesseracts of Odin and of Galois.

From an obituary for Helen Nicoll, author
of a popular series of British children's books—

"They feature Meg, a witch whose spells
always seem to go wrong, her cat Mog,
and their friend Owl." 

For some (very loosely) related concepts that
have been referred to in this journal, see…

Meg,  Mog,  and Owl.

See, too, "Kids grow up" (Feb. 13, 2012).

» more

This, together with this morning's post involving the squares 16 and 9,
suggests a review of Conceit at Harvard (October 25, 2006), which
contains the following figure involving the squares 16, 9, and 25—

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Verlyn Klinkenborg in yesterday's online New York Times—

"Even metaphors — the best ones anyway —
are literal-minded. But that’s a story for another day."

Another day: May 18, 2010—

Part I: At Pomona College

"Writer-in-Residence Verlyn Klinkenborg '74
Writes Essay on Graduation for New York Times"
— Pomona College news item, May 18, 2010, by
   Laura Tiffany

Part II: In this journal

Note that the geometric diamond in the screenshot above
is not blue but black.

See also Pomona College under the topic Defining Form
in this journal.

Today's previous post, "For Odin's Day," discussed 
a mathematical object, the tesseract, from a strictly
narrative point of view.

In honor of George Balanchine, Odin might yield the
floor this evening to Apollo.

From a piece in today's online New York Times  titled
"How a God Finds Art (the Abridged Version)"—

"… the newness at the heart of this story,
in which art is happening for the first time…."

Some related art—

and, more recently—

This more recent figure is from Ian Stewart's 1996 revision 
of a 1941 classic, What Is Mathematics? , by Richard Courant
and Herbert Robbins.

Apollo might discuss with Socrates how the confused slave boy
of Plato's Meno  would react to Stewart's remark that

"The number of copies required to double an
 object's size depends on its dimension."

Apollo might also note an application of Socrates' Meno  diagram
to the tesseract of this afternoon's Odin post—

.

(Continued from Walpurgisnacht 2012)

Wikipedia article on functional decomposition—

"Outside of purely mathematical considerations,
perhaps the greatest value of functional decomposition
is the insight it provides into the structure of the world."

Certainly this is true for the sort of decomposition
known as harmonic analysis .

It is not, however, true of my own decomposition theorem,
which deals only with structures made up of at most four
different sorts of elementary parts.

But my own approach has at least some poetic value.

See the four elements of the Greeks in (for instance)
Eliot's Four Quartets  and in Auden's For the Time Being .

"In ancient Greece, 9 was the number of the Muses,
patron goddesses of the arts. They were the daughters
of Mnemosyne ('memory'), the source
of imagination, which in turn is the carrier of archetypal,
elementary ideas to artistic realization in the field
of space-time. The number 9, that is to say, relates
traditionally to the Great Goddess of Many Names
(Devi, Inanna, Ishtar, Astarte, Artemis, Venus, etc.),
as matrix of the cosmic process, whether in the
macrocosm or in a microcosmic field of manifestation."

— Joseph Campbell in The Inner Reaches of Outer Space ,
      first published in 1986

(Continued from In Memoriam (Aug. 22), Chapman's Homer (Aug. 23),
and this morning's Colorful Tale)

An informative, but undated, critique of the late Marvin W. Meyer
by April D. DeConick at the website of the Society of Biblical Literature
appeared in more popular form in an earlier New York Times
op-ed piece, "Gospel Truth," dated Dec. 1, 2007.

A check, in accord with Jungian synchronicity, of this  journal
on that date yields a quotation from Plato's Phaedrus  —

"The soul or animate being has the care of the inanimate."

Related verses from T. S. Eliot's Four Quartets —

"The detail of the pattern is movement."

"So we moved, and they, in a formal pattern."

Some background from pure mathematics (what the late
William P. Thurston called "the theory of formal patterns")—

The Animated Diamond Theorem.

On the middle initial of the Cary Grant character
in yesterday's post Summer Reading—

Click image for further details.

"The concept of nothingness follows Roger Thornhill throughout North by Northwest , first as another identity imposes itself upon him and later as circumstances force him to run from Vandamm as well as the police. When Eve asks him what the 'O' in 'ROT' stands for, Thornhill can only answer 'nothing.' His middle initial's lack of meaning connects well to the overall theme of the human self as possibly nothing." —Hitchcock and Identity, by Emily Pilgrim

Related material— Elementary Finite Geometry (Aug. 1).

See, too, a post for Holy Cross Day in 2002.

On Robert A. Heinlein's novel Glory Road—

"Glory Road  (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road  was influenced by the debut of the science fiction television series Doctor Who  on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."

— Todd, Tesseract article at exampleproblems.com

From the same exampleproblems.com article—

"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."

For further details, see today's new page on vertex adjacency at finitegeometry.org.

(Continued)

A Necessary Truth—

James Singer, "A Theorem in Finite Projective Geometry
and Some Applications to Number Theory," Transactions
of the American Mathematical Society  43 (1938), 377-385.

A Contingent Truth—

Singer Tony Martin reportedly died Friday evening, July 27, 2012.

In his memory, some references to a "Singer 7-Cycle."

See also this journal 7 years prior to Martin's death.

(Continued)

Eric M. Friedlander, President of the
American Mathematical Society (AMS),
in the March 2011 AMS Notices —

"I think the best thing the AMS does by far is the Notices .
It could easily be in all doctors’ and dentists’ offices."

Notices : "Really?"

Friedlander: "It could be."

Related material from this journal:

“Is it safe?"

— Annals of Art Education: 
     Geometry and Death

For art collector Herbert Vogel,
who reportedly died today

Philip Kennicott in The Washington Post , July 3, 2009— 

"The Vogels help allay deep cultural fears
within the art world— fears that art is elitist,
or some kind of confidence game,
or not a serious endeavor (a fear that has
dogged art since at least the time of Plato)."

Some related material from finitegeometry.org,
offered without comment—

(Continued)

For Pete Rustan, space recon expert, who died on June 28—

(Click to enlarge.)

See also Galois vs. Rubik and Group Theory Template.

Heidegger, "Hölderlin and the Essence of Poetry,"
translated by Douglas Scott, in Existence and Being  ,
Regnery, 1949, pp. 291-316—

See also Hexagram 36.

Notes on Groups and Geometry, 1978-1986,
is now available at the Internet Archive.

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

This post was suggested by a link from a post
 in this journal seven years ago yesterday—

“Is the language of thought
 any more than a dream?“

— Rimbaud

Yes.

References here yesterday to Epiphany 
and to Chinese logic suggest two observations—

First, a political interpretation of the number "64.89" 
from Monday's Shanghai stock market index
yielded the date  6/4 in 1989—

Second, an interpretation of 64.89 as the number  64 in 1989
(on the Feast of the Epiphany, January 6)—

Note of January 6, 1989 showing 
the 64 hexagrams in the von Franz style

No connection of the number  64 with the date  6/4 is implied.

Wikipedia— 

"Clowns to the left of me, jokers to the right"

 — Songwriter who died on January 4, 2011.

Online NY Times  on the date of the songwriter's death—

"A version of this review appeared in print
on January 4, 2011, on page C6 of the New York edition." 

REVIEW

"The philosopher Hubert Dreyfus and his former student
Sean Dorrance Kelly have a story to tell, and it is not
a pretty tale for us moderns. Ours is an age of nihilism,
they say, meaning not so much that we have nothing
in which to believe, but that we don’t know how to choose
among the various things to which we might commit
ourselves. Looking down from their perches at Berkeley
and Harvard, they see the 'human indecision that
plagues us all.'"

For an application of the excluded-middle law, see
Non-Euclidean Blocks and Deep Play.

Violators of the law may have trouble* distinguishing
between "Euclidean" and "non-Euclidean" phenomena
because their definition of the latter is too narrow,
based only on examples that are historically well known.

See the Non-Euclidean Blocks  footnote.

* Followers  of the excluded-middle law will avoid such
trouble by noting that "non-Euclidean" should mean
simply "not  Euclidean in some  way "— not  necessarily
in a way contradicting Euclid's parallel postulate.

But see Wikipedia's defense of the standard, illogical,
usage of the phrase "non-Euclidean."

Postscript—

Tertium Datur

"Here I am, stuck in the middle with you."

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

The Digital Wittgenstein:

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

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

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

The latter contrast is exemplified in Pilate Goes to Kindergarten.

* An analogy, as it were, between  analogies.

From the NY Times  philosophy column "The Stone" 
yesterday at 5 PM—

Timothy Williamson, Wykeham Professor of Logic at Oxford,
claims that all the theorems of mathematics

"… are ultimately derived from a few simple axioms
by chains of logical reasoning, some of them
hundreds of pages long…."

Williamson gives as an example recent (1986-1995)
work on Fermat's conjecture.

He does not, however, cite any axioms or "chains of
logical reasoning" in support of his claim that 
a proof of Fermat's conjecture can be so derived.

Here is a chain of reasoning that forms a crucial part
of recent arguments for the truth of Fermat's conjecture—

K. A. Ribet, "On modular representations of Gal(Q̄/Q)
arising from modular forms," Invent. Math. 100 (1990), 431-476.

Whether this chain of reasoning is in fact logical  is no easy question.
It is not the sort of argument easily reduced to a series of purely
logical symbol-strings that could be checked by a computer.

Few mathematicians, even now, can follow each step
in the longer chain of reasoning that led to a June 1993 claim
that Fermat's conjecture is true. 

Williamson is not a mathematician, and his view of
Fermat's conjecture as a proven fact is clearly based
not on logic, but on faith.

Last evening's Geometry of the Dance discussed
a book on the Norwegian mathematician
Niels Henrik Abel. The post dealt with the group
S₄ of 24 permutations of a 4-element set.

                                                    "In that open field
If you do not come too close, if you do not come too close,
On a summer midnight, you can hear the music…."

— The dance in Four Quartets

For a summer midnight related to the group S₄,
see Midnight in Oslo from last August.

"At the still point…." — T. S. Eliot

"…a dance results." — Marie-Louise von Franz

Peter Pesic uses a dance metaphor to explain
finite group theory, with permutations of four elements
represented by symmetries of a tetrahedron—

For a different approach to the dance metaphor, see
the dance in Four Quartets and Poetry's Bones.

                                             In that open field
If you do not come too close, if you do not come too close,
On a summer midnight, you can hear the music
Of the weak pipe and the little drum
And see them dancing around the bonfire
The association of man and woman
In daunsinge, signifying matrimonie—
A dignified and commodiois sacrament.
Two and two, necessarye coniunction….

This post continues the April 9 post
commemorating Élie Cartan's birthday.

That post mentioned triality .
Here is John Baez reviewing
On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry
by John H. Conway and Derek A. Smith
(A.K. Peters, Ltd., 2003)—

"In this context, triality manifests itself
as the symmetry that cyclically permutes
the Hurwitz integers  i , j ,  and k ."

Related material— Quaternion Acts in this journal
as well as Finite Geometry and Physical Space.

Now online—

Notes on Groups and Geometry,
1978-1986, by Steven H. Cullinane

PDF, 3.4 MB.

(Continued from Part I and Part II.)

The paper excerpted below supplies some badly needed technical
background for the Wikipedia article on functional decomposition.

The preprint above gives the precise definitions and technical references
that are completely absent from Wikipedia's Functional decomposition.

For some related material on 4×4 arrays like those in the above figure
see Decomposition Part I and Geometry of the 4×4 Square.

Compare and contrast

1. The following excerpt from Wikipedia—

2. A webpage subtitled "Function Decomposition Over a Finite Field."

Related material—

Decomposition and Jews Telling Stories.

A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.

From that essay—

"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."

Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—

It seems some clarification is in order.

Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.

Scientific American  on the concept of digital space—

"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."

Wikipedia on the concept of quantization—

Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….

For a purely mathematical  approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.

The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .

Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to 
physics; Galois geometry has not. The cited
webpage describes the interplay of both  sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.

The 3×3×3 Galois Cube

Backstory— The Talented, from April 26 last year,
and Atlas Shrugged, from April 27 last year.

"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."

— G. H. Hardy, A Mathematician's Apology

In Geometry and the Imagination , Hilbert and Cohn-Vossen
describe the Brianchon-Pascal configuration of 9 points
and 9 lines, with 3 points on each line and 3 lines through
each point, as being "the most important configuration of all geometry."

The Brianchon-Pascal configuration is also known as the Pappus configuration—

Some background …

"The Theorem of Pappus: A Bridge Between Algebra and Geometry"
Elena Anne Marchisotto
The American Mathematical Monthly
Vol. 109, No. 6 (Jun. – Jul., 2002), pp. 497-516

“Now (lens“

— Finnegans Wake , Book II,
Episode 2, page 293

Background—

• The Great Hunger,
• the facetious NY Times ‘s
  ‘Hunger Games’ Parenting,
• and (Feb. 13 and 14, 2012)—
  Why Now? and Notable Transitions.

"Is Space Digital?" 

— Cover story, Scientific American  magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment at his physics weblog today

See also 

• Digital Space, 
• Galois vs. Rubik,
• Solomon's Cube, and
• Cuber.

* See yesterday's Steiner's Systems.