Log24

Monday, November 4, 2019

Trudeau Revisited

Filed under: General — Tags: — m759 @ 2:24 PM

Previously in Log24:  Trudeau and the Story Theory  of Truth.

More-recent remarks by Trudeau

Bible Stories for Skeptics

Review
With a bit of a twinkle in his eye, Richard Trudeau—a skeptic, retired Unitarian Universalist minister, Harvard Divinity School grad and (though he doesn't use the word) humanist—removes the supernaturalism from some of the common stories in the Judeo-Christian Bible for the edification of non-scholars. He links them to history as known to archeologists and serious historians and tries to salvage some things of value in the collection of diverse materials in the book…. 
— Edd Doerr, book reviewer for The Unitarian Universalist Humanist Association

About the Author
Richard Trudeau is minister emeritus of the Unitarian Universalist Church of Weymouth, Massachusetts. He holds a Master of Divinity degree with concentration in biblical studies from Harvard Divinity School. He is also professor emeritus of Stonehill College in Easton, Massachusetts, where he specialized in the history of mathematics, the philosophy of science and the history of astronomy. His previous books include Universalism 101 and The Non-Euclidean Revolution.

Product details
File Size: 474 KB
Print Length: 166 pages
Page Numbers Source ISBN: 1493688642
Publisher: Chad Brown Publishing (July 6, 2014)

Log24 on the above publication date — July 6, 2014 —

Friday, August 30, 2019

The Coxeter Aleph

Filed under: General — Tags: — m759 @ 4:21 AM

(Continued)

The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.

A meditation, with apologies to Coleridge:

In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.

A related video —

Beware, beware, her flashing eyes, her floating hair:

Set design —

As opposed to block design

Monday, February 20, 2012

Coxeter and the Relativity Problem

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

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"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, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"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, The Classical Groups , Princeton University Press, 1946, p. 16

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of  the primitive root λ or 10 are

λ = 10,  λ2 = 21,  λ3 = 22,  λ4 = 02,
λ5 = 20,  λ6 = 12,  λ7 = 11,  λ8 = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"

http://www.log24.com/log/pix12/120220-CoxeterFig10.jpg

Coxeter's Figure 10 yields...

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The Aleph

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

Friday, January 7, 2011

Coxeter and the Aleph

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

In a nutshell —

Epigraph to "The Aleph," a 1945 story by Borges:

O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2

http://www.log24.com/log/pix11/110107-BorgesElAleph.jpg

The story in book form, 1949

A 2006 biography of geometer H.S.M. Coxeter:

http://www.log24.com/log/pix11/110107-KingOfInfiniteSpace-Sm.jpg

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Related material: Group Actions, 1984-2009.

Tuesday, November 5, 2019

Non-Woo

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

A followup to earlier posts on Trudeau vs. Euclid —

Geometry from July 6, 2014:

IMAGE- Concepts of Space

Monday, November 4, 2019

Science Woo

Filed under: General — Tags: — m759 @ 4:03 PM

Sounds like a story by Optimus Prime

"Before time began, there was the Cube."

Euclid Revisited

Filed under: General — Tags: — m759 @ 6:48 AM

The version of Euclid I.47 in the previous post
suggests a work from a recent Oslo gallery show:

 

As Above, So Below*

Filed under: General — Tags: — m759 @ 5:43 AM
 

Braucht´s noch Text?

* An "established rule of law
across occult writings.
"

Friday, July 5, 2019

The Motive for Metaphor

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

"János Bolyai was a nineteenth-century mathematician who
set the stage for the field  of non-Euclidean geometry."

Transylvania Now , October 26, 2018

 

From  Coxeter and the Relativity Problem

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

Wallace Stevens, "The Motive for Metaphor"

Friday, April 19, 2019

Pleasantly Discursive Day

Filed under: General — m759 @ 9:05 AM

" 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

Greek Cross, adapted from painting by Ad Reinhardt

Wednesday, January 2, 2019

“Pleasantly Discursive” Continues.

Filed under: General — m759 @ 12:21 PM

From this journal on December 13th, 2016

" 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

Also on December 13th, 2016 —

Friday, September 28, 2018

ART WARS Midrash

Filed under: General — m759 @ 9:00 AM

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

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

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks
on the "story theory" of truth as opposed to the "diamond theory"
of truth in The Non-Euclidean Revolution  (1987)

The deaths of Roth and Grünbaum on September 14th,
The Feast of the Holy Cross, along with Douthat's column
today titled "Only the Truth Can Save Us Now," suggest a
review of

Elements of Number Theory, by Vinogradov .

Tuesday, January 9, 2018

Unpleasantly Discursive

Filed under: General — m759 @ 10:12 PM

Background for the remarks of Koen Thas in the previous post —
Schumacher and Westmoreland, "Modal Quantum Theory" (2010).

Related material —

" 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

The whole  truth may require an unpleasantly  discursive treatment.

Example —

1. The reported death on Friday, Jan. 5, 2018, of a dancer
     closely associated with George Balanchine

2. This journal on Friday, Jan. 5, 2018:

3. Illustration from a search related to the above dancer:

4. "Per Mare Per Terras" — Clan slogan above, illustrated with
     what looks like a cross-dagger.

    "Unsheathe your dagger definitions." — James Joyce.

5. Discursive remarks on quantum theory by the above
    Schumacher and Westmoreland:

6. "How much story do you want?" — George Balanchine

Wednesday, August 9, 2017

Implosion

Filed under: General,Geometry — Tags: , — m759 @ 9:48 PM

For those whose only interest in mathematics
is as a path to the occult —

See also Coxeter's Aleph.

Monday, July 17, 2017

Athens Meets Jerusalem . . .

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

At the Googleplex .

For those whose only interest in higher mathematics
is as a path to the occult

Plato's Diamond and the Hebrew letter Aleph —

          

and some related (if only graphically) mathematics —

Click the above image for some related purely mathematical  remarks.

Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That  cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

These are Rothko's Swamps .

See a Log24 search for related meditations.

For all three topics combined, see Coxeter

" 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

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Tuesday, December 13, 2016

The Thirteenth Novel

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

John Updike on Don DeLillo's thirteenth novel, Cosmopolis

" DeLillo’s post-Christian search for 'an order at some deep level'
has brought him to global computerization:
'the zero-oneness of the world, the digital imperative . . . . ' "

The New Yorker , issue dated March 31, 2003

On that date ….

Related remark —

" 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

Friday, May 6, 2016

ART WARS continues…

Filed under: General — m759 @ 11:00 AM

"Again, in spite of that, we call this Friday good."
— T. S. Eliot, Four Quartets

From this journal on Orthodox Good Friday, 2016,
an image from New Scientist  on St. Andrew's Day, 2015 —

From an old Dick Tracy strip —

See also meditations from this year's un -Orthodox Good Friday
in a Tennessee weblog and in this  journal

" 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

Friday, March 25, 2016

Pleasantly Discursive

Filed under: General — m759 @ 10:00 AM

Toronto geometer H.S.M. Coxeter, introducing a book by Unitarian minister
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

Another such treatment

"Of course, it will surprise no one to find low standards
of intellectual honesty on the Tonight Show.

But we find a less trivial example if we enter the
hallowed halls of Harvard University. . . ."

— Neal Koblitz, "Mathematics as Propaganda"

Less pleasantly and less discursively —

"Funny how annoying a little prick can be."
— The late Garry Shandling

Sunday, October 18, 2015

Coordinatization Problem

Filed under: General,Geometry — Tags: — m759 @ 1:06 AM

There are various ways to coordinatize a 3×3 array
(the Chinese "Holy Field'). Here are some —

See  Cullinane,  Coxeter,  and  Knight tour.

Saturday, November 16, 2013

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

IMAGE- The 'atomic square' in Lee Sallows's article 'The Lost Theorem'

Yes. See

The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =

(1,1) (1,0) (1,2)
(0,1) (0,0) (0,2)
(2,1) (2,0) (2,2)

Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.

Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and  
all four broken diagonals (with arithmetic modulo 3).

For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.

Tuesday, August 13, 2013

The Story of N

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

(Continued from this morning)

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The above stylized "N," based on
an 8-cycle in the 9-element Galois field
GF(9), may also be read as
an Aleph.

Graphic designers may prefer a simpler,
bolder version:

Tuesday, May 14, 2013

Raiders of the Lost Aleph

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

See Coxeter + Aleph in this journal.

Epigraph to "The Aleph," a 1945 story by Borges:

"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2

Sunday, December 9, 2012

Eve’s Menorah

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

"Now the serpent was more subtle
than any beast of the field…."
Genesis 3:1

"“The serpent’s eyes shine
As he wraps around the vine….”
Don Henley

"Nine is a vine."
Folk rhyme

Part I

Part II

Part III

Halloween 2005

The image “http://log24.com/log/pix03/030109-gridsmall.gif” cannot be displayed, because it contains errors.

Click images for some background.

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.

Counterexample—

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

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics
http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

Monday, October 10, 2011

10/10

Filed under: General — Tags: — m759 @ 10:00 AM

See last year's Day of the Tetraktys.

Those who prefer Hebrew to Greek may consult Coxeter and the Aleph.

See also last midnight's The Aleph as well as Saturday morning's
An Ordinary Evening in Hartford and Saturday evening's
For Whom the Bell (with material from March 20, 2011).

For connoisseurs of synchronicity, there is …

THE LAST CONCERT

Cached from http://mrpianotoday.com/tourdates.htm —
The last concert of Roger Williams — March 20, 2011 —

   March 20

"Roger Williams" In Concert,
The Legendary Piano Man!!
Roger Williams & his Band
(Sierra Ballroom)
7:30-9:00pm

Palm Desert, CA    

Background music… Theme from "Somewhere in Time"

Thursday, May 19, 2011

The Aleph, the Lottery, and the Eightfold Way

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

Three links with a Borges flavor—

Related material

The 236 in yesterday evening's NY lottery may be
 viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.

A new illustration for that topic—

http://www.log24.com/log/pix11A/110519-8-3-Configuration.jpg

This shows a reconcilation of the triples described by Sloane
 in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.

Note that  the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of  Sloane's 83 configuration—

http://www.log24.com/log/pix11A/110519-SloaneDesign.jpg

The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.

Wednesday, May 18, 2011

Minimalist Icon

Filed under: General,Geometry — Tags: — m759 @ 6:48 AM

The source of the mysterious generic
3×3 favicon with one green cell

http://www.log24.com/log/pix11A/110518-GenericFavicon.jpg

— has been identified.

For minimalists, here is a purer 3×3 matrix favicon—

http://www.log24.com/log/pix11A/110518-3x3FaviconURL.jpg

This may, if one likes, be viewed as the "nothing"
present at the Creation.  See Jim Holt on physics.

See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.

Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube

http://www.log24.com/log/pix11A/110509-SpideySuperStories39Sm.jpg

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

http://www.log24.com/log/pix11A/110510-GaloisAleph.GIF

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Friday, April 22, 2011

Romancing the Hyperspace

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

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Image-- examples from Galois affine geometry

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"–

"… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry."

Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory."

Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds.

* "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

Friday, January 7, 2011

Ayn Sof

Filed under: General — Tags: , — m759 @ 7:26 PM

(A continuation of this morning's Coxeter and the Aleph)

"You've got to pick up every stitch… Must be the season of the witch."
Donovan song at the end of Nicole Kidman's "To Die For"

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics

http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

"As is well known, the Aleph is the first letter of the Hebrew alphabet.
Its use for the strange sphere in my story may not be accidental.
For the Kabbala, the letter stands for the En Soph ,
the pure and boundless godhead; it is also said that it takes
the shape of a man pointing to both heaven and earth, in order to show
that the lower world is the map and mirror of the higher; for Cantor's
Mengenlehre , it is the symbol of transfinite numbers,
of which any part is as great as the whole."

— Borges, "The Aleph"

From WorldLingo.com

Ein Sof

Ein Soph or Ayn Sof (Hebrew  אין סוף, literally "without end", denoting "boundlessness" and/or "nothingness"), is a Kabbalistic term that usually refers to an abstract state of existence preceding God's Creation of the limited universe. This Ein Sof , typically referred to figuratively as the "light of Ein Sof " ("Or Ein Sof "), is the most fundamental emanation manifested by God. The Ein Sof  is the material basis of Creation that, when focused, restricted, and filtered through the sefirot , results in the created, dynamic universe.
….

Cultural impact

Mathematician Georg Cantor labeled different sizes of infinity using the Aleph. The smallest size of infinity is aleph-null (0), the second size is aleph-one (1), etc. One theory about why Cantor chose to use the aleph is because it is the first letter of Ein-Sof. (See Aleph number)

"Infinite Jest… now stands as the principal contender
for what serious literature can aspire to
in the late twentieth and early twenty-first centuries."

All Things Shining, a work of pop philosophy published January 4th

http://www.log24.com/log/pix10B/101231-AllThingsShining-Cover.jpg

"You're gonna need a bigger boat." — Roy Scheider in "Jaws"

"We're gonna need more holy water." — "Season of the Witch," a film opening tonight

See also, with respect to David Foster Wallace, infinity, nihilism,
and the above reading of "Ayn Sof" as "nothingness,"
the quotations compiled as "Is Nothing Sacred?"

Saturday, November 20, 2010

Search

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

An Epic Search for Truth

— Subtitle of Logicomix , a work reviewed in the December 2010 Notices of the American Mathematical Society  (see previous post).

Some future historian of mathematics may contrast the lurid cover of the December 2010 Notices

http://www.log24.com/log/pix10B/101119-AMSnoticesThumb.jpg

Excerpts from Logicomix

with the 1979 cover found in a somewhat less epic search —

http://www.log24.com/log/pix10B/101120-79T-A37-500w.jpg

Larger view of Google snippet —

http://www.log24.com/log/pix10B/101120-79T-A37-snippet.jpg

For some purely mathematical background, see Finite Geometry of the Square and Cube.

For some background related to searches for truth, see "Coxeter + Trudeau" in this journal.

Thursday, February 18, 2010

Theories: An Outline

Filed under: General,Geometry — 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.)

Sunday, September 27, 2009

Sunday September 27, 2009

Filed under: General,Geometry — m759 @ 3:00 AM
A Pleasantly
Discursive Treatment

In memory of Unitarian
minister Forrest Church,
 dead at 61 on Thursday:

NY Times Sept. 27, 2009, obituaries, featuring Unitarian minister Forrest Church

Unitarian Universalist Origins: Our Historic Faith

“In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history.”

Gravity’s Rainbow–

“For every kind of vampire, there is a kind of cross.”

Unitarian minister Richard Trudeau

“… I called the belief that

(1) Diamonds– informative, certain truths about the world– exist

the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that

(2) The theorems of Euclidean geometry are diamonds….

As the news about non-Euclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.

Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.

Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The Non-Euclidean Revolution, p. 255

H. S. M. Coxeter, 1987, introduction to Trudeau’s book

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…

A kind
 of cross:

Diamond formed by four diagonally-divided two-color squares

See both
Theme and
Variations
and some more
poetic remarks,

Mirror-Play
 of the Fourfold.

* As recent Log24 entries have pointed out, diamond theory (in the original 1976 sense) is a type of non-Euclidean geometry, since finite geometry is not Euclidean geometry– and is, therefore, non-Euclidean, in the strictest sense (though not according to popular usage).

Friday, April 10, 2009

Friday April 10, 2009

Filed under: General,Geometry — m759 @ 8:00 AM

Pilate Goes
to Kindergarten

“There is a pleasantly discursive
 treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”

— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
 remarks on the “Story Theory
 of truth as opposed to the
Diamond Theory” of truth in
 The Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.

Simplifying:

“Can a finite geometry describe physical space?”

Simplifying further:

“Yes. VideThe Eightfold Cube.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube

Friday, January 30, 2009

Friday January 30, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
Two-Part Invention

This journal on
October 8, 2008,
at noon:

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

— H. S. M. Coxeter, introduction to Richard J. Trudeau’s remarks on the “story theory” of truth as opposed to the “diamond theory” of truth in The Non-Euclidean Revolution

Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees.

Insidehighered.com on
the same day, October 8, 2008,
at 12:45 PM EDT

“Future readers may consider Updike our era’s Mozart; Mozart was once written off as a too-prolific composer of ‘charming nothings,’ and some speak of Updike that way.”

— Comment by BPJ

“Birthday, death-day–
 what day is not both?”
John Updike

Updike died on January 27.
On the same date,
Mozart was born.

Requiem

Mr. Best entered,
tall, young, mild, light.
He bore in his hand
with grace a notebook,
new, large, clean, bright.

— James Joyce, Ulysses,
Shakespeare and Company,
Paris, 1922, page 178

Related material:

Dec. 5, 2004 and

Inscribed carpenter's square

Jan. 27-29, 2009

Wednesday, October 8, 2008

Wednesday October 8, 2008

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

Serious Numbers

A Yom Kippur
Meditation

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

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

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution

Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–

"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"

(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)

Richard J. Trudeau on the 'Story Theory' of truth

Excerpt from
The Non-Euclidean Revolution

What does this have to do with numbers?

Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.

Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.

For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.

Thursday, March 6, 2008

Thursday March 6, 2008

Filed under: General,Geometry — m759 @ 12:00 PM
This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA’s Icosahedron Society.

Royal Road

“The historical road
from the Platonic solids
to the finite simple groups
is well known.”

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that “there is no royal road to geometry.” The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, “Map-Colouring Problems,” in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. “the king of geometry.”

See also Log24, May 21, 2007.

A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M24.

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”

— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M24):

“Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code.” [Here I is the identity matrix and J is the matrix of all 1’s.]

Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube

and
Jewel in the Crown

“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?'”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
“story theory” of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.


* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Tuesday, April 25, 2006

Tuesday April 25, 2006

Filed under: General,Geometry — m759 @ 3:09 PM

“There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to
Richard J. Trudeau’s remarks on
the “Story Theory” of truth
as opposed to
the “Diamond Theory” of truth
in The Non-Euclidean Revolution

A Serious Position

“‘Teitelbaum,’ in German,
is ‘date palm.'”
Generations, Jan. 2003   

“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”

Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church
)

From Nextbook.org, “a gateway to Jewish literature, culture, and ideas”:

NEW BOOKS: 02.16.05
Proofs and Paradoxes
Alfred Teitelbaum changed his name to Tarski in the early 20s, the same time he changed religions, but when the Germans invaded his native Poland, the mathematician was in California, where he remained. His “great achievement was his audacious assault on the notion of truth,” says Martin Davis, focusing on the semantics and syntax of scientific language. Alfred Tarski: Life and Logic, co-written by a former student, Solomon Feferman, offers “remarkably intimate information,” such as abusive teaching and “extensive amorous involvements.”

From Wikipedia, an unsigned story:

“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”

The image “http://www.log24.com/log/pix06/060425-Tarski.jpg” cannot be displayed, because it contains errors.

Alfred Tarski

The image “http://www.log24.com/log/pix06/060424-Crimson2.jpg” cannot be displayed, because it contains errors.

See also
 
The Crimson Passion.

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — m759 @ 4:04 PM
An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

The image “http://www.log24.com/log/pix05B/051120-NonEuclideanRev.jpg” cannot be displayed, because it contains errors.

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to  the “Diamond Theory” of truth in The Non-Euclidean Revolution

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”

Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

 In 1490, The White Knight
 (Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
 a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

Wednesday, September 28, 2005

Wednesday September 28, 2005

Filed under: General,Geometry — m759 @ 4:26 AM
Mathematical Narrative,
continued:

There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
“What is truth?”

— H. S. M. Coxeter, introduction to
Richard J. Trudeau’s
The Non-Euclidean Revolution

“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….The happy metaphor is Morris Kline’s in Mathematics in Western Culture (Oxford, 1953), p. 430.”

— Richard J. Trudeau,
   The Non-Euclidean Revolution,
   Birkhauser Boston,
   1987, pages 114 and 117

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call ‘models of reality’). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture.”

  — Richard J. Trudeau,
     The Non-Euclidean Revolution,
     Birkhauser Boston,
     1987, pages 256 and 259

An example of
the story theory of truth:

The image “http://www.log24.com/log/pix05B/050925-Proof1.jpg”  cannot be displayed, because it contains errors.

Actress Gwyneth Paltrow (“Proof”) was apparently born on either Sept. 27, 1972, or Sept. 28, 1972.   Google searches yield  “about 193” results for the 27th and “about 610” for the 28th.

Those who believe in the “story theory” of truth may therefore want to wish her a happy birthday today.  Those who do not may prefer the contents of yesterday’s entry, from Paltrow’s other birthday.

Friday, August 19, 2005

Friday August 19, 2005

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

Mathematics and Narrative
continued

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

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to  the "Diamond Theory" of truth " in The Non-Euclidean Revolution

"I had an epiphany: I thought 'Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?'"

An organizer of last month's "Mathematics and Narrative" conference

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*…."

Richard J. Trudeau in The Non-Euclidean Revolution

"'Deniers' of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others."

— Jim Holt in this week's New Yorker magazine.  Click on the box below.

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

* Many stripes

   "What disciplines were represented at the meeting?"
   "Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it."

 

An organizer of last month's "Mathematics and Narrative" conference

Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: — m759 @ 12:07 AM

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets.  It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

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


H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

 
The "horse's head" figure above is from a note I wrote on this date 18 years ago.  The following journal entry from April 4, 2003, gives some details:
 

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.

Monday, April 7, 2003

Monday April 7, 2003

Filed under: General,Geometry — Tags: — m759 @ 1:17 PM

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

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


— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:


 

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