Log24

Sunday, October 29, 2017

Rivals

Filed under: Uncategorized — m759 @ 11:00 PM

The passage from Lewis Carroll's Euclid and His Modern Rivals 
in the previous post suggests two illustrations —

Click the Trudeau book for related Log24 posts.

Friday, December 16, 2016

Memory, History, Geometry

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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

Saturday, March 5, 2016

The Looming

Filed under: Uncategorized — m759 @ 3:14 PM

Material related to the title:

  • From the post Edifice (March 1, 2016) —

"Euclid's edifice loomed in my consciousness
as a marvel among sciences, unique in its clarity
and unquestionable validity."
—Richard J. Trudeau in 
   The Non-Euclidean Revolution  (1986)

Tuesday, March 1, 2016

Edifice

Filed under: Uncategorized — m759 @ 12:00 PM

"Euclid's edifice loomed in my consciousness as a marvel among
sciences, unique in its clarity and unquestionable validity."
—Richard J. Trudeau in The Non-Euclidean Revolution  (1986)

On 'The Public Square,' from 'Edgar Allan Poe, Wallace Stevens, and the Poetics of American Privacy'

See also Edifice in this journal and last night's architectural post.

Monday, October 27, 2014

Revolutions in Geometry

Filed under: Uncategorized — m759 @ 9:00 AM

A post in honor of Évariste Galois (25 October 1811 – 31 May 1832)

From a book by Richard J. Trudeau titled The Non-Euclidean Revolution

See also “non-Euclidean” in this journal.

One might argue that Galois geometry, a field ignored by Trudeau,
is also “non-Euclidean,” and  (for those who like rhetoric) revolutionary.

Monday, October 21, 2013

Edifice Complex

Filed under: Uncategorized — m759 @ 8:00 PM

New! Improved!

"Euclid's edifice loomed in my consciousness 
as a marvel among sciences, unique in its
clarity and unquestionable validity." 
—Richard J. Trudeau in
   The Non-Euclidean Revolution  (First published in 1986)

Readers of this journal will be aware that Springer's new page
advertising Trudeau's book, pictured above, is a bait-and-switch
operation. In the chapter advertised, Trudeau promotes what he
calls "the Diamond Theory of Truth" as a setup for his real goal,
which he calls "the Story Theory of Truth."

For an earlier use of the phrase "Diamond Theory" in
connection with geometry, see a publication from 1977.

Friday, July 5, 2013

Self-Evident

Filed under: Uncategorized — m759 @ 4:30 AM

Google sidebar for Richard J. Trudeau's 'The Non-Euclidean Revolution'

Trudeau is a sophist.

Wertheim, on the other hand

Thursday, November 1, 2012

Theories of Truth

Filed under: Uncategorized — Tags: — m759 @ 7:20 PM

A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The Non-Euclidean Revolution

The Story Theory of Truth:

"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."

New Yorker  profile of tesseract
     author Madeleine L'Engle

The Diamond Theory of Truth:

(Click image for some background.)

Spaces as Hypercubes

See also the links on a webpage at finitegeometry.org.

Sunday, January 29, 2012

Declarations

Filed under: Uncategorized — m759 @ 6:00 PM

Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.

See Gowers and Cameron.

My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).

A 1976 document relevant to the concerns in the Trudeau book—

Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.

They apply to finite  geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations  
of mathematics.

"The hand of the creative artist, laid upon the major premise,
 rocks the foundations of the world." — Dorothy Sayers

Wednesday, August 10, 2011

Objectivity

Filed under: Uncategorized — m759 @ 12:25 PM

From math16.com

Quotations on Realism
and the Problem of Universals:

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268

"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.

"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

Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is non-Euclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000

The correct statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)— 

From The Search for Invariants (June 20, 2011):

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

http://www.log24.com/log/pix11B/110810-MajaLovrenovBio.jpg

Related material from Sunday's New York Times  travel section—

"Exhibit A is certainly Ljubljana…."

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: Uncategorized — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Friday, April 22, 2011

Romancing the Hyperspace

Filed under: Uncategorized — 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 .

Tuesday, May 25, 2010

Sisteen

Filed under: Uncategorized — Tags: , — m759 @ 9:57 AM

“Nuvoletta in her lightdress, spunn of sisteen shimmers,
was looking down on them, leaning over the bannistars….

Fuvver, that Skand, he was up in Norwood’s sokaparlour….”

Finnegans Wake

To counteract the darkness of today’s 2:01 AM entry—

Part I— Artist Josefine Lyche describes her methods

A “Internet and hard work”
B “Books, both fiction and theory”

Part II I, too, now rely mostly on the Internet for material. However, like Lyche, I have Plan B— books.

Where I happen to be now, there are piles of them. Here is the pile nearest to hand, from top to bottom.

(The books are in no particular order, and put in the same pile for no particular reason.)

  1. Philip Rieff— Sacred Order/Social Order, Vol. I: My Life Among the Deathworks
  2. Dennis L. Weeks— Steps Toward Salvation: An Examination of Coinherence and Substitution in the Seven Novels of Charles Williams
  3. Erwin Panofsky— Idea: A Concept in Art Theory
  4. Max Picard— The World of Silence
  5. Walter J. Ong, S. J.— Hopkins, the Self, and God
  6. Richard Robinson— Definition
  7. X. J. Kennedy and Dana Gioia, eds.— An Introduction to Poetry
  8. Richard J. Trudeau— The Non-Euclidean Revolution
  9. William T. Noon, S. J.— Joyce and Aquinas
  10. Munro Leaf— Four-and-Twenty Watchbirds
  11. Jane Scovell— Oona: Living in the Shadows
  12. Charles Williams— The Figure of Beatrice
  13. Francis L. Fennell, ed.— The Fine Delight: Centenary Essays on the Poetry of Gerard Manley Hopkins
  14. Hilary Putnam— Renewing Philosophy
  15. Paul Tillich— On the Boundary
  16. C. S. Lewis— George MacDonald

Lyche probably could easily make her own list of what Joyce might call “sisteen shimmers.”

Sunday, May 9, 2010

Today’s Sermon

Filed under: Uncategorized — m759 @ 11:00 AM

School Book Depository
(Revisited)

Image-- Heath, 'A History of Greek Mathematics'
Pro-Truth
Image-- Trudeau, 'The Non-Euclidean Revolution'
Pro-Lies

Thursday, April 22, 2010

Mere Geometry

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

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

Stanford Encyclopedia of Philosophy

Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”

A non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

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.

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.)

Sunday, September 27, 2009

Sunday September 27, 2009

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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.

Tuesday, April 25, 2006

Tuesday April 25, 2006

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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

Saturday, June 14, 2003

Saturday June 14, 2003

Filed under: Uncategorized — m759 @ 5:00 PM

Indiana Jones
and the Hidden Coffer

In memory of Bernard Williams,

Oxford philosopher, who died Tuesday, June 10, 2003. 

“…in… Truth and Truthfulness [September, 2002], he sought to speak plainly, and took on the post-modern, politically correct notion that truth is merely relative…”

— Christopher Lehmann-Haupt

“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….

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….”

— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987

Today is the feast day of Saint Jorge Luis Borges (b. Buenos Aires, August 24, 1899 – d. Geneva, June 14, 1986).

From Borges’s “The Aleph“:

“The Faithful who gather at the mosque of Amr, in Cairo, are acquainted with the fact that the entire universe lies inside one of the stone pillars that ring its central court…. The mosque dates from the seventh century; the pillars come from other temples of pre-Islamic religions…. Does this Aleph exist in the heart of a stone?”

(“Los fieles que concurren a la mezquita de Amr, en el Cairo, saben muy bien que el universo está en el interior de una de las columnas de piedra que rodean el patio central…. la mezquita data del siglo VII; las columnas proceden de otros templos de religiones anteislámicas…. ¿Existe ese Aleph en lo íntimo de una piedra?”)

From The Hunchback of Notre Dame:

Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar.*

* A coffer of great richness
In a pillar’s heart they found,
Within it lay new banners,
With figures to astound.

See also the figures obtained by coloring and permuting parts of the above religious symbol.

Lena Olin and Harrison Ford
in “Hollywood Homicide

Monday, April 28, 2003

Monday April 28, 2003

Filed under: Uncategorized — 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: Uncategorized — 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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