Log24

Tuesday, August 30, 2005

Tuesday August 30, 2005

Filed under: Uncategorized — m759 @ 5:20 PM
Mathematics and Narrative,
Continued:


The Happy Ending Problem

From Google News this afternoon–

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See also the previous entry.

Monday, August 29, 2005

Monday August 29, 2005

Filed under: Uncategorized — Tags: — m759 @ 4:00 PM
VALE

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George and Esther Szekeres

From the weblog of
David Michael Brown, Jr.:
 

Date:     Sun, 28 Aug 2005
             12:30:40 -0400
From:    Alf van der Poorten AM
           
Subject: Vale George Szekeres and
             Esther Klein Szekeres

Members of the Number Theory List will be sad to learn that George and Esther Szekeres both died this morning.  George, 94, had been quite ill for the last 2-3 days, barely conscious, and died first at 06:30.  Esther, 95, died a half hour later.

Both George Szekeres and Esther Klein will be recalled by number theorists as members of the group of young Hungarian mathematicians of the 1930s including Turan and Erdos.  George and Esther's coming to Australia in the late 40s played an important role in the invigoration of Australian Mathematics.  George was also an expert in group theory and relativity; he was my PhD supervisor.

Emeritus Professor
Alf van der Poorten AM
Centre for Number Theory Research
1 Bimbil Place, Killara NSW

 

Related material:

AVE

3:09 PM EDT Thursday, Aug. 25, 2005:
 

  "Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one." 

 

  "A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?"

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   "It is demonstrated that space-time should possess a discrete structure on Planck scales."

   — Peter Szekeres, abstract of Discrete Space-Time

Peter Szekeres is the son of George and Esther Szekeres.
 

ATQUE

"At present, such relationships can at best be heuristically described in terms that invoke some notion of an 'intelligent user standing outside the system.'"

Gian-Carlo Rota in Indiscrete Thoughts, p. 152
 

Related material:
High Concept and
Nothing Nothings (Again).

Saturday, August 27, 2005

Saturday August 27, 2005

Filed under: Uncategorized — m759 @ 10:00 PM

Diamond Theorem Revisited

This evening I wrote a revised version of my 1979 “diamond theorem” abstract.

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: Uncategorized — m759 @ 3:09 PM
Analogical
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

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

Another representation of
the 24-cell
:

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
,”
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:

[…]

“I’d like to wrap up with a few small comments about last Week.  There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’.  The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”

[…]

Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.

NDE


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics
:


Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London

Disclaimer:

The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Wednesday, August 24, 2005

Wednesday August 24, 2005

Filed under: Uncategorized — m759 @ 12:00 AM

High Concept, continued:

“In the beginning there was nothing.
 And God said, ‘Let there be light!’
 And there was still nothing,
 but now you could see it.

— Jim Holt, Big-Bang Theology,
    Slate‘s “High Concept” department

Related material:

  1. On the phrase “verbum mentis”
  2. From Satan’s Rhetoric, by Armando Maggi
    (University of Chicago Press, 2001):
Page 110:

“In chapter I I explained that devils first and foremost exist as semioticians of the world’s signs.  Devils solely live in their interpretations, in their destructive syllogisms.  As Visconti puts it, devils speak the idiom of the mind.37  …. The exorcist’s healing voice states that Satan has always been absent from the world, that his disturbing and unclear manifestations in the possessed person’s physicality are really nonexistent occurrences, nothing but disturbances of the mind, since evil itself is a lack of being.”   

Footnote 37, page 110:

“It is necessary to distinguish the devils’ ‘language of the mind’ and Augustine’s verbum mentis (word of the mind), as he theorizes it first of all in On the Trinity (book 15).  The devils’ language of the mind disturbs the subject’s internal and preverbal discourse.”

Tuesday, August 23, 2005

Tuesday August 23, 2005

Filed under: Uncategorized — m759 @ 1:06 PM
1:06:55
PM

The image “http://www.log24.com/images/IChing/hexagram55.gif” cannot be displayed, because it contains errors.

I had an epiphany.”
Apostolos Doxiadis   

Related material:
Log24 March 11:
Lucas Promises a
Darker Star Wars

Tuesday August 23, 2005

Filed under: Uncategorized — m759 @ 12:00 PM

High Concept*

“Concept (scholastics’ verbum mentis)–
 theological analogy of Son’s procession
 as Verbum Patris, 111-12″
 — index to Joyce and Aquinas,
 by William T. Noon, S.J.,
Yale University Press 1957,
 second printing 1963, page 162

“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”

— Jim Holt, Big-Bang Theology, Slate‘s “High Concept” department

Related material:

Nothing Ventured,
The God-Shaped Hole, and
Is Nothing Sacred?

 * See also John O’Callaghan, Thomistic Realism and the Linguistic Turn: Toward a More Perfect Form of Existence, (University of Notre Dame Press, 2003) and Joshua P. Hochschild, “Does Mental Language Imply Mental Representationalism? The Case of Aquinas’s Verbum Mentis,” Proceedings of the Society for Medieval Logic and Metaphysics, Volume 4, 2004 (pdf), pp. 12-17.

Tuesday August 23, 2005

Filed under: Uncategorized — m759 @ 2:45 AM
Quarter to Three

You’d never know it,
But buddy, I’m a kind of poet…

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Monday, August 22, 2005

Monday August 22, 2005

Filed under: Uncategorized — m759 @ 4:07 PM
The Hole

Part I: Mathematics and Narrative

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Apostolos Doxiadis on last month’s conference on “mathematics and narrative”–

Doxiadis is describing how talks by two noted mathematicians were related to

    “… a sense of a ‘general theory bubbling up’ at the meeting… a general theory of the deeper relationship of mathematics to narrative….

Doxiadis says both talks had “a big hole in the middle.”  

    “Both began by saying something like: ‘I believe there is an important connection between story and mathematical thinking. So, my talk has two parts.  [In one part] I’ll tell you a few things about proofs.  [And in the other part] I’ll tell you about stories.’ …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other.”
  “And the hole?”
  This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’  But the similarity was not clearly explained: the hole, you see, the ‘miracle.’  Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one.”

Part II: Possible Worlds

“At times, bullshit can only be countered with superior bullshit.”
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    “The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges’ seminal short story El jardín de senderos que se bifurcan (“The Garden of Forking Paths“) is an early example of many worlds in fiction.”

Il faut cultiver notre jardin.“– Voltaire 

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole
 

Part III: Modal Theology

  “‘What is this Stone?’ Chloe asked….
  ‘…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'”

  — Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)

“The lapis was thought of as a unity and therefore often stands for the prima materia in general.”

  — Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)

“Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe.”

  — The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)

“We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes)).

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

“The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being….”

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Saturday, August 20, 2005

Saturday August 20, 2005

Filed under: Uncategorized — m759 @ 2:07 PM
Truth vs. Bullshit

Background:
For an essay on the above topic
from this week’s 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.

Representing truth:

The image “http://www.log24.com/log/pix05B/050820-Goldstein.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein

Representing bullshit:

The image “http://www.log24.com/log/pix05B/050820-Doxiadis.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis

Goldstein’s truth:

Gödel was a Platonist who believed in objective truth.

See Rothstein’s review of Goldstein’s new book Incompleteness.

Doxiadis’s bullshit:

Gödel, along with Darwin, Marx, Nietzsche, Freud, Einstein, and Heisenberg, destroyed a tradition of certainty that began with Plato and Euclid.

“Examples are the stained-glass
windows of knowledge.” — Nabokov

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

Thursday, August 18, 2005

Thursday August 18, 2005

Filed under: Uncategorized — m759 @ 1:06 PM
Final Arrangements,
continued:

The image “http://www.log24.com/log/pix05B/050818-Sinatra.jpg” cannot be displayed, because it contains errors.

“Mr. Deutsch, a jaunty, elegant figure, was known as Ardie to his friends. Those friends included the composer Frank Loesser, who was his roommate for a time, and Frank Sinatra, with whom he spent many a marathon weekend of whiskey, pasta and golf in Palm Springs.”

Todd S. Purdum in today’s New York Times

Thursday August 18, 2005

Filed under: Uncategorized — m759 @ 12:48 AM
Sermon for
World Youth Day
 

(Cologne, Aug. 16-21, 2005)

“And the light shineth in darkness;
and the darkness comprehended it not.”
— The Gospel according to St. John,
Chapter 1, Verse 5 

Part I: The Light

The Shining of May 29
and
Diamond Theory

Part II: The Darkness

Mathematics and Narrative
and
Reply to My Fan Mail

Wednesday, August 17, 2005

Wednesday August 17, 2005

Filed under: Uncategorized — m759 @ 12:00 PM
At Cologne

The image “http://www.log24.com/log/pix05B/050817-Ludi.jpg” cannot be displayed, because it contains errors.

    “The Game was at first nothing more than a witty method for developing memory and ingenuity among students and musicians.
     The inventor, Bastian Perrot of Calw… found that the pupils at the Cologne Seminary had a rather elaborate game they used to play. One would call out, in the standardized abbreviations of their science, motifs or initial bars of classical compositions, whereupon the other had to respond with the continuation of the piece, or better still with a higher or lower voice, a contrasting theme, and so forth. It was an exercise in memory and improvisation quite similar to the sort of thing probably in vogue among the ardent pupils of counterpoint in the days of Schütz, Pachelbel, and Bach….
     Bastian Perrot… constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors….”

Hermann Hesse at The Glass Bead Game Defined

Tuesday, August 16, 2005

Tuesday August 16, 2005

Filed under: Uncategorized — m759 @ 12:07 PM

Narrative and Latin Squares

From The Independent, 15 August 2005:

“Millions of people now enjoy Sudoku puzzles. Forget the pseudo-Japanese baloney: sudoku grids are a version of the Latin Square created by the great Swiss mathematician Leonhard Euler in the late 18th century.”

The Independent
was discussing the conference on “Mathematics and Narrative” at Mykonos in July.

From the Wikipedia article on Latin squares:

“The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version).

The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of “orthogonal squares” (Diamond Theory, 1976) or “orthogonal matrices”– orthogonal, that is, in a combinatorial, not a linear-algebra sense (A. E. Brouwer, 1991).”

This last paragraph, added to Wikipedia on Aug. 14,  may or may not survive the critics there.

Saturday, August 13, 2005

Saturday August 13, 2005

Filed under: Uncategorized — m759 @ 2:00 PM

Kaleidoscope, continued:

Austere Geometry

From Noel Gray, The Kaleidoscope: Shake, Rattle, and Roll:

“… what we will be considering is how the ongoing production of meaning can generate a tremor in the stability of the initial theoretical frame of this instrument; a frame informed by geometry’s long tradition of privileging the conceptual ground over and above its visual manifestation.  And to consider also how the possibility of a seemingly unproblematic correspondence between the ground and its extrapolation, between geometric theory and its applied images, is intimately dependent upon the control of the truth status ascribed to the image by the generative theory.  This status in traditional geometry has been consistently understood as that of the graphic ancilla– a maieutic force, in the Socratic sense of that term– an ancilla to lawful principles; principles that have, traditionally speaking, their primary expression in the purity of geometric idealities.*  It follows that the possibility of installing a tremor in this tradition by understanding the kaleidoscope’s images as announcing more than the mere subordination to geometry’s theory– yet an announcement that is still in a sense able to leave in place this self-same tradition– such a possibility must duly excite our attention and interest.

* I refer here to Plato’s utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave.”

See also

Noel Gray, Ph.D. thesis, U. of Sydney, Dept. of Art History and Theory, 1994:

“The Image of Geometry: Persistence qua Austerity– Cacography and The Truth to Space.”

Saturday August 13, 2005

Filed under: Uncategorized — m759 @ 12:04 PM

Kaleidoscope, continued:

In Derrida’s Defense

The previous entry quoted an attack on Jacques Derrida for ignoring the “kaleidoscope” metaphor of Claude Levi-Strauss.  Here is a quote by Derrida himself:

“The time for reflection is also the chance for turning back on the very conditions of reflection, in all the senses of that word, as if with the help of an optical device one could finally see sight, could not only view the natural landscape, the city, the bridge and the abyss, but could view viewing. (1983:19)

— Derrida, J. (1983) ‘The Principle of Reason: The University in the Eyes of its Pupils’, Diacritics 13.3: 3-20.”

The above quotation comes from Simon Wortham,  who thinks the “optical device” of Derrida is a mirror.  The same quotation appears in Desiring Dualisms at thispublicaddress.com, where the “optical device” is interpreted as a kaleidoscope.

Derrida’s “optical device” may (for university pupils desperately seeking an essay topic) be compared with Joyce’s “collideorscape.”  For a different connection with Derrida, see The ‘Collideorscape’ as Différance.

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: Uncategorized — m759 @ 8:16 AM

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

Tuesday, August 9, 2005

Tuesday August 9, 2005

Filed under: Uncategorized — m759 @ 5:01 PM

Kaleidoscope

A new web page simplifies the Diamond 16 Puzzle and relates the resulting “kaleidoscope” to Hesse’s Bead Game.

Sunday, August 7, 2005

Sunday August 7, 2005

Filed under: Uncategorized — m759 @ 12:12 PM
Religious Symbolism
at Harvard

Photo adapted from
today’s online New York Times:

The image “http://www.log24.com/log/pix05B/050807-Howard.jpg” cannot be displayed, because it contains errors.

For details, see
The Da Vinci Code
and Symbology at Harvard
.

Sunday August 7, 2005

Filed under: Uncategorized — m759 @ 7:20 AM
Presbyterian Justice

News from today’s New York Times:

The Rev. Dr. Theodore Alexander Gill Sr., a Presbyterian theologian, a philosophy teacher, and an influential provost emeritus of John Jay College of Criminal Justice in Manhattan, died at 85 on June 10 in Princeton.  In retirement from John Jay, The Rev. Dr. Gill was theologian in residence at Nassau Presbyterian Church in Princeton.

In memory of The Rev. Dr. Gill:

Religious Symbolism at Princeton
    (on Nassau Presbyterian Church),
Pro-Semitism
    (on number theory at Princeton),
For the Mad Musicians of Princeton,
     (on Schroeder and Bernstein),
Movie Date and its preceding entries
   (on Princeton’s St. John von Neumann),
Why Me?
   (for Princeton theologian Elaine Pagels),
Notes on Literary and Philosophical Puzzles
   (Princeton’s John Nash as Ya Ya Fontana), and
Go Tigers!
   (for the Princeton Evangelical Fellowship).

For a more conventional memorial, see

The image “http://www.log24.com/log/pix05B/050807-SFTS-Logo.gif” cannot be displayed, because it contains errors.

the obituary from

San Francisco Theological Seminary.

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: Uncategorized — m759 @ 1:25 PM

The Fugue

   "True joy is a profound remembering, and true grief is the same.
    Thus it was, when the dust storm that had snatched Cal up finally died, and he opened his eyes to see the Fugue spread out before him, he felt as though the few fragile moments of epiphany he'd tasted in his twenty-six years– tasted but always lost– were here redeemed and wed. He'd grasped fragments of this delight before. Heard rumour of it in the womb-dream and the dream of love; known it in lullabies. But never, until now, the whole, the thing entire.
    It would be, he idly thought, a fine time to die.
    And a finer time still to live, with so much laid out before him."

— Clive Barker,
Weaveworld,
 Book Two:
The Fugue

From Monday:

Weaveworld,
Book Three:
Out of the
Empty Quarter

"The wheels of its body rolled,
the visible mathematics
    of its essence turning on itself…."

From Friday:

The image “http://www.log24.com/log/pix05B/050806-Square.bmp” cannot be displayed, because it contains errors.

  For the meaning
of this picture, see
Geometry of the
4×4 Square.

For graphic designs
based on this geometry,
see Theme and Variations
and Diamond Theory.

For these designs in the
context of a Bach fugue,
see Timothy A. Smith's
essay (pdf) on

Fugue No. 21 in B-Flat Major
from Book II of
The Well-Tempered Clavier
by Johann Sebastian Bach.

Smith also offers a
Shockwave movie
that uses diamond theory
to illustrate this fugue.

Saturday August 6, 2005

Filed under: Uncategorized — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

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In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Saturday August 6, 2005

Filed under: Uncategorized — m759 @ 8:15 AM
Music for the
Feast of the
Transfiguration

Jesus hits like
an atom bomb.

The image “http://www.log24.com/log/pix05/050312-AtomBomb.jpg” cannot be displayed, because it contains errors.
Click on picture
for a sound clip.

Friday, August 5, 2005

Friday August 5, 2005

Filed under: Uncategorized — Tags: , — m759 @ 4:23 PM

For Sir Alec

From Elegance:

"Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday…."

— Bernard Holland, page C12,
    The New York Times,
    Monday, May 20, 1996.

Holland was pondering the identity of the Juilliard String Quartet, which had just given a series of concerts celebrating its fiftieth anniversary.

"Elegant"

— Page one,
    The New York Times,
    Monday, August 7, 2000.
 
The Times was describing the work of Sir Alec Guinness, who died on 8/5/00.

An example of the Holland name problem:

Monday, August 1, 2005 — Visible Mathematics:

    "Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
   '
"Empty" is a misnomer…  the Rub' al-Khali contains many hidden riches.'"

Friday, August 5, 2005 —  

The image “http://www.log24.com/log/pix05B/050805-Rag.jpg” cannot be displayed, because it contains errors.

Related material:

Geometry for Prince Harry

Friday August 5, 2005

Filed under: Uncategorized — Tags: , — m759 @ 1:16 PM

Abel's Birthday

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Click on picture
for details.

Thursday, August 4, 2005

Thursday August 4, 2005

Filed under: Uncategorized — m759 @ 1:00 PM
Visible Mathematics, continued

Today’s mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Related material: Visualizing GL(2,p)

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Wednesday, August 3, 2005

Wednesday August 3, 2005

Filed under: Uncategorized — m759 @ 2:02 PM

Epiphany Term

“In Epiphany Term, 1942, C.S. Lewis delivered the Riddell Memorial Lectures… in….  the University of Durham….  He delivered three lectures entitled ‘Men without Chests,’ ‘The Way,’ and ‘The Abolition of Man.’  In them he set out to attack and confute what he saw as the errors of his age. He started by quoting some fashionable lunacy from an educationalists’ textbook, from which he developed a general attack on moral subjectivism.  In his second lecture he argued against various contemporary isms, which purported to replace traditional objective morality.  His final lecture, ‘The Abolition of Man,’ which also provided the title of the book published the following year, was a sustained attack on hard-line scientific anti-humanism. The intervening fifty years have largely vindicated Lewis.”

— J. R. Lucas, The Restoration of Man

See also Log24,
Epiphany 2003.
 

Tuesday, August 2, 2005

Tuesday August 2, 2005

Filed under: Uncategorized — m759 @ 10:18 AM
Austere

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

Pictured:
Modal Theology

By SALAH NASRAWI
The Associated Press

Tuesday, August 2, 2005  9:50 AM EDT

RIYADH, Saudi Arabia — Muslim leaders and Saudi princes bade farewell to King Fahd on Tuesday, saying prayers in a packed Riyadh mosque and then burying him in an unmarked desert grave in keeping with the kingdom’s austere version of Islam.

Tuesday August 2, 2005

Filed under: Uncategorized — Tags: — m759 @ 7:00 AM
Today's birthday:
Peter O'Toole

"What is it, Major Lawrence,
 that attracts you personally
 to the desert?"

"It's clean."

Visible Mathematics,
continued —

From May 18:

Lindbergh's Eden

"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."

— Anne Morrow Lindbergh,
Earth Shine, p. xii
 

 
On Beauty
 
"Beauty is the proper conformity
of the parts to one another
and to the whole."

— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974

Related material:

The Eightfold Cube

The Eightfold Cube

(in Arabic, ka'b)

and

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Tuesday August 2, 2005

Filed under: Uncategorized — Tags: — m759 @ 5:24 AM

Final Arrangements, continued
 

Kismet

From yesterday's Log24
Clive Barker's Weaveworld:

Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.

If I stand starry-eyed
That's a danger in paradise
For mortals who stand beside
  An angel like you.

Robert Wright and George Forrest

The image “http://www.log24.com/log/pix05B/050802-NYTobits2.jpg” cannot be displayed, because it contains errors.
 

Tuesday August 2, 2005

Filed under: Uncategorized — m759 @ 12:00 AM
50 Years Ago
on this date, poet
Wallace Stevens died.

Memorial: at the
Wallace Stevens
Concordance,
enter center.

Monday, August 1, 2005

Monday August 1, 2005

Filed under: Uncategorized — m759 @ 11:00 PM

Final Arrangements, continued

Ready for Her Closeup

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From today’s New York Times:

“BERLIN, July 31 – Willem F. Duisenberg, the blunt-spoken Dutch central banker who oversaw the introduction of the euro as the first president of the European Central Bank, was found dead on Sunday in a swimming pool at his villa in the south of France….”

The image “http://www.log24.com/log/pix05B/050801-Closeup.jpg” cannot be displayed, because it contains errors.

Flashback to July 29,
Anatomy of a Death

Monday August 1, 2005

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM
Visible Mathematics

    "Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
     '
"Empty" is a misnomer…  the Rub' al-Khali contains many hidden riches.'"

Maps from the Sky,
   Saudi Aramco World, March/April 1995

From Weaveworld

Book Three:
Out of the Empty Quarter,
 by Clive Barker, 1987:


… As a child he'd learned the names of all the angels and archangels by heart: and among the mighty, Uriel was of the mightiest.  The archangel of salvation: called by some the flame of God…. What had he done, stepping into the presence of such power?  This was Uriel, of the principalities….
    Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.

 

"The serpent's eyes shine
As he wraps around the vine
In the Garden of Allah."

Don Henley, 1995  
 

Monday August 1, 2005

Filed under: Uncategorized — m759 @ 9:05 AM
In memory of filmmaker

Kayo Hatta

From Google News,
8:15 AM EDT today:

The image “http://www.log24.com/log/pix05B/050801-Fahd2.jpg” cannot be displayed, because it contains errors.

A brief film:

The image “http://www.log24.com/log/pix05B/050801-Fahd5.jpg” cannot be displayed, because it contains errors.

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