Log24

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: , — m759 @ 10:01 am

It is well known that the seven (22 + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (32 + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (42 + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field  GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

Number and Time, by Marie-Louise von Franz

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G20160 in Mitchell 1910,  LF(3,22) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
   of the Finite Projective Plane PG(2,22),"
   Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
   the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

Sunday, February 21, 2010

Reflections

Filed under: General,Geometry — Tags: , , — m759 @ 12:06 pm

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

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

Finite fields

This section requires expansion.

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

Related material:

"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and

"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"

by Ascher Wagner, U. of Birmingham, received 27 July 1977

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

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

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

Clearly the eightfold cube is a counterexample.

Tuesday, February 16, 2010

Mysteries of Faith

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

From today's NY Times

http://www.log24.com/log/pix10/100216-NYTobits.jpg

Obituaries for mystery authors
Ralph McInerny and Dick Francis

From the date (Jan. 29) of McInerny's death–

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"

Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson

From the date (Feb. 14) of Francis's death–

2x2x2 cube

The EIghtfold Cube

The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.

This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.

For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
  and the death of Coxeter–

Putting Descartes Before Dehors

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

For a more Protestant meditation,
see The Cross of Descartes

Descartes

Descartes's Cross

"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke

For further details, click on
the image below–

Quine and Derrida at Notre Dame Philosophical Reviews

Notre Dame Philosophical Reviews

Friday, October 2, 2009

Friday October 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 6:00 am
Edge on Heptads

Part I: Dye on Edge

“Summary:
….we obtain various orbits of partitions of quadrics over GF(2a) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell’s heptagons for the Klein quadric in PG(5,2).”

Introduction:
In 1910 Conwell… produced his heptagons in PG(5,2) associated with the Klein quadric K whose points represent the lines of PG(3,2)…. Edge… constructed the 8 heptads of complexes in PG(3,2) directly. Both he and Conwell used their 8 objects to establish geometrically the isomorphisms SL(4,2)=A8 and O6(2)=S8 where O6(2) is the group of K….”

— “Partitions and Their Stabilizers for Line Complexes and Quadrics,” by R.H. Dye, Annali di Matematica Pura ed Applicata, Volume 114, Number 1, December 1977, pp. 173-194

Part II: Edge on Heptads

The Geometry of the Linear Fractional Group LF(4,2),” by W.L. Edge, Proc. London Math Soc., Volume s3-4, No. 1, 1954, pp. 317-342. See the historical remarks on the first page.

Note added by Edge in proof:
“Since this paper was finished I have found one by G. M. Conwell: Annals of Mathematics (2) 11 (1910), 60-76….”

Some context:

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Wednesday, August 19, 2009

Wednesday August 19, 2009

Filed under: General,Geometry — Tags: — m759 @ 7:00 pm
From Visualizing GL(2,p)
to Visualizing GL(2,Z)

A note from 1985 leads,
via today’s earlier entry,
to an article from 1993:

Visualizing Toral Automorphisms-- The opening paragraphs
See also
 Arnold’s Cat Map.

Sunday, May 17, 2009

Sunday May 17, 2009

Filed under: General,Geometry — Tags: , — m759 @ 7:59 am
Design Theory

Laura A. Smit, Calvin College, "Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams"–

"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."

 

Hans Ludwig de Vries, "On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16)," Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–

"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."

1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).

4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).

Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–

"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."

Olli Pottonen,  'Classification of Steiner Quadruple Systems'

"Simplify, simplify."
Thoreau

"Beauty is bound up
with symmetry."
Weyl

Sixteen points in a 4x4 array

Pottonen's thesis is
 dated Nov. 16, 2005.

For some remarks on
images and theology,
see Log24 on that date.

Click on the above image
 for some further details.

Saturday, April 4, 2009

Saturday April 4, 2009

Filed under: General,Geometry — Tags: , — m759 @ 8:00 am
Annual Tribute to
The Eight

Katherine Neville's 'The Eight,' edition with knight on cover, on her April 4 birthday

Other knight figures:

Knight figures in finite geometry (Singer 7-cycles in the 3-space over GF(2) by Cullinane, 1985, and Curtis, 1987)

The knight logo at the SpringerLink site

Click on the SpringerLink
knight for a free copy
(pdf, 1.2 mb) of
the following paper
dealing with the geometry
underlying the R.T. Curtis
knight figures above:

Springer description of 1970 paper on Mathieu-group geometry by Wilbur Jonsson of McGill U.

Context:

Literature and Chess and
Sporadic Group References

Details:

 

Adapted (for HTML) from the opening paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups M22, M23, M24…"–

"[A]… uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely

PGL(4,2)~PSL(4,2)~SL(4,2)~A8,
PSp(4,2)~Sp(4,2)~S6,

where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II).

The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau…."

References

4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910).

5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954).

7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.

Saturday, March 28, 2009

Saturday March 28, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:07 pm

The Rest
of the Story

Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.

The rest of the story:
 

The Revelation Game
(continued from 7/26, 2008)

 
Lotteries
on Reba's
birthday,
2009
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
No belief,
no revelation

726
Revelation
without belief

378
Evening
(Belief)
Belief without
revelation

006
Belief and
revelation

091

Interpretations of the evening numbers–

The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.
 

Will: Do you like apples?     
Clark: Yeah.                       
Will: Well, I got her number.
 How do you like them apples?

— "Good Will Hunting

Kate Beckinsale in 'Underworld: Evolution'

The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–

Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–
 

Page 91 of 'The Innermost Kernel' by Suzanne Gieser, Springer 2005

The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."
 

A particularly
marked circle
 from March 15:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

The circle above is
marked with a version of
the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

The square above is marked
with a graphic design
related to the four-diamond
figure of Jung's Aion.

Sunday, March 1, 2009

Sunday March 1, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 am

Solomon's Cube
continued

"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."

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

"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia

"One has O+(6) ≅ S8, the symmetric group of order 8! …."

 — "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.

"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Saturday, February 28, 2009

Saturday February 28, 2009

Filed under: General,Geometry — Tags: — m759 @ 8:00 am
Mathematics
and Narrative

continued

Narrative:

xxx

Mathematics:

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference, (July 2000), Springer, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Mathematics:

"Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and O6 + (2)."

J. Sutherland Frame in "Orthogonal Groups over GF(2) and Related Graphs," Springer Lecture Notes in Mathematics vol. 642, Theory and Applications of Graphs (Proceedings, Michigan, May 11–15, 1976), edited by Y. Alavi and D. R. Lick, pp. 174-185

"One has O+(6) ≅ S8, the symmetric group of order 8!…."

— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint. This paper gives some context in superstring theory for the following work of Frame:

[F1] J.S. Frame, The classes and representations of the group of 27 lines and 28 bitangents, Annali
di Mathematica Pura ed Applicata, 32 (1951) 83–119.
[F2] J.S. Frame, Some characters of orthogonal groups over the field of two elements, In: Proc. of the
Second Inter. Conf. on the Theory of Groups, Lecture Notes in Math., Vol. 372, pp. 298–314,
Springer, 1974.
[F3] J. S. Frame, Degree polynomials for the orthogonal groups over GF(2), C. R. Math. Rep. Acad.
Sci. Canada 2 (1980) 253–258.

Friday, October 24, 2008

Friday October 24, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 8:08 am

The Cube Space” is a name given to the eightfold cube in a vulgarized mathematics text, Discrete Mathematics: Elementary and Beyond, by Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3-space over the 2-element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the less-obvious generation of the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.

 

Saturday, August 16, 2008

Saturday August 16, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:00 am

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Finite Structure
 on a Book Cover:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.


Walsh Series
states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

Sunday, June 1, 2008

Sunday June 1, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 2:14 pm
Yet Another
Cartoon Graveyard

The conclusion of yesterday’s commentary on the May 30-31 Pennsylvania Lottery numbers:

Thomas Pynchon, Gravity’s Rainbow:

“The fear balloons again inside his brain. It will not be kept down with a simple Fuck You…. A smell, a forbidden room, at the bottom edge of his memory. He can’t see it, can’t make it out. Doesn’t want to. It is allied with the Worst Thing.

He knows what the smell has to be: though according to these papers it would have been too early for it, though he has never come across any of the stuff among the daytime coordinates of his life, still, down here, back here in the warm dark, among early shapes where the clocks and calendars don’t mean too much, he knows that’s what haunting him now will prove to be the smell of Imipolex G.

Then there’s this recent dream he is afraid of having again. He was in his old room, back home. A summer afternoon of lilacs and bees and

286”

What are we to make of this enigmatic 286? (No fair peeking at page 287.)

One possible meaning, given The Archivists claim that “existence is infinitely cross-referenced”–

Page 286 of Ernest G. Schachtel, Metamorphosis: On the Conflict of Human Development and the Psychology of Creativity (first published in 1959), Hillsdale NJ and London, The Analytic Press, 2001 (chapter– “On Memory and Childhood Amnesia”):

“Both Freud and Proust speak of the autobiographical [my italics] memory, and it is only with regard to this memory that the striking phenomenon of childhood amnesia and the less obvious difficulty of recovering any past experience may be observed.”

The concluding “summer afternoon of lilacs and bees” suggests that 286 may also be a chance allusion to the golden afternoon of Disney’s Alice in Wonderland. (Cf. St. Sarah’s Day, 2008)

Some may find the Disney afternoon charming; others may see it as yet another of Paul Simon’s dreaded cartoon graveyards.

More tastefully, there is poem 286 in the 1919 Oxford Book of English Verse– “Love.”

For a midrash on this poem, see Simone Weil, who became acquainted with the poem by chance:

“I always prefer saying chance rather than Providence.”

— Simone Weil, letter of about May 15, 1942

Weil’s brother André might prefer Providence (source of the Bulletin of the American Mathematical Society.)

Andre Weil and his sister Simone, summer of 1922(Photo from Providence)

 

Related material:


Log24, December 20, 2003–
White, Geometric, and Eternal

A description in Gravity’s Rainbow of prewar Berlin as “white and geometric”  suggested, in combination with a reference elsewhere to “the eternal,” a citation of the following illustration of the concept “white, geometric, and eternal”–

For more on the mathematical significance of this figure, see (for instance) Happy Birthday, Hassler Whitney, and Combinatorics of Coxeter Groups, by Anders Björner and Francesco Brenti, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005.

This book is reviewed in the current issue (July 2008) of the above-mentioned Providence Bulletin.

The review in the Bulletin discusses reflection groups in continuous spaces.

For a more elementary approach, see Reflection Groups in Finite Geometry and Knight Moves: The Relativity Theory of Kindergarten Blocks.

See also a commentary on
the phrase “as a little child.”

Saturday, February 23, 2008

Saturday February 23, 2008

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

"An acute study of the links
between word and fact"
Nina daVinci Nichols

 
Thanks to a Virginia reader for a reminder:
 
Virginia /391062427/item.html? 2/22/2008 7:37 PM
 
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.

The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure–

Change

The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Thursday, August 9, 2007

Thursday August 9, 2007

Filed under: General — Tags: — m759 @ 12:00 pm
“Serious numbers  
will always be heard.”

— Paul Simon

(See St. Luke’s Day, 2005.)  


Bulletin of the American Mathematical Society
,
Volume 31, Number 1, July 1994, Pages 1-14

Selberg’s Conjectures
and Artin L-Functions
(pdf)

M. Ram Murty

Introduction

In its comprehensive form, an identity between an automorphic L-function and a “motivic” L-function is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of Shimura-Taniyama that every elliptic curve over Q is “modular” is certainly the most intriguing reciprocity conjecture of our time. The “Himalayan peaks” that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent. In either case, a long journey is predicted…. At the 1989 Amalfi meeting, Selberg [S] announced a series of conjectures which looks like another approach to the summit. Alas, neither path seems the easier climb….

[S] A. Selberg, Old and new
      conjectures and results
      about a class of Dirichlet series,
      Collected Papers, Volume II,
      Springer-Verlag, 1991, pp. 47-63.

Zentralblatt MATH Database
on the above Selberg paper:

“These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989…. There are various stimulating conjectures (which are related to several other conjectures like the Sato-Tate conjecture, Langlands conjectures, Riemann conjecture…)…. Concluding remark of the author: ‘A more complete account with proofs is under preparation and will in time appear elsewhere.'”

Related material: Previous entry.

Tuesday, August 7, 2007

Tuesday August 7, 2007

Filed under: General — Tags: — m759 @ 8:00 am
The Horse Whisperer

Scarlett Johansson and friend in The Horse Whisperer

Scarlett Johansson and friend
in “The Horse Whisperer” (1998)

Thanks to University Diaries (Aug. 6) for the following:

“‘The University of Sydney has ordered an independent review into allegations that the dean of the Conservatorium of Music hired a horse whisperer to conduct management workshops.’ [Are you, like UD, a bit vague on exactly what a horse whisperer is? And are you having trouble figuring out what a horse whisperer would have to offer a management workshop? But then, what exactly is a management workshop? Read on.]”

For some background on horse whispering and management workshops, see IABC Steal Sheet, March 2004.

Related material:

The recent Log24 entries

University Diaries:
“God, isn’t there already
enough crap in this story?”

See also Log24,
Dec. 10, 2003:

Putting Descartes Before Dehors

      

“Descartes déclare que
c’est en moi, non hors de moi,
en moi, non dans le monde,
que je pourrais voir
si quelque chose existe
hors de moi.”

ATRIUM, Philosophie     

For further details,
see ART WARS.

Tuesday, July 17, 2007

Tuesday July 17, 2007

Filed under: General — Tags: — m759 @ 7:00 am
Habeas Corpus
 
The Hex Witch of Seldom,
by Nancy Springer:

Hex Witch of Seldom - Excerpt on squares of breadT

Log24 on 9/11, 2003
:

Here is a rhetorical exercise
for Jesuits that James Joyce
might appreciate:

Discuss Bobbi’s “little squares”
of bread as the Body of Christ.
Formulate, using Santayana’s
criteria, a definition of beauty
that includes this sacrament.

Wednesday, June 20, 2007

Wednesday June 20, 2007

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

Kernel

Mathematical Reviews citation:

MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9

A quote from MR at Amazon.com:

"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
 

An excerpt:

 

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

I do not yet know what Gieser means by "the innermost kernel." The following is my version of a "kernel" of sorts– a diagram well-known to students of anthropologist Claude Levi-Strauss and art theorist Rosalind Krauss:

The four-group is also known as the Vierergruppe or Klein group.  It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)

Related material:

The "chessboard" of
   Nov. 7, 2006
(as revised Nov. 7, 2012)–

I Ching chessboard. Previous version replaced on Nov. 7, 2012, by original 1989 chessboard arrangement

I Ching chessboard

None of this material really has much to do with the history of physics, except for its relation to the life and thought of physicist Wolfgang Pauli— the "Mephistopheles" of the new book Faust in Copenhagen. (See previous entry.)

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
 

Sunday, November 19, 2006

Sunday November 19, 2006

Filed under: General — Tags: — m759 @ 2:02 pm

Signature

From AP’s “Today in History,” Nov. 19, 2006:

Today’s birthdays: … Actress-director Jodie Foster is 44….

Thought for Today: “My theology, briefly, is that the universe was dictated but not signed.” –[Attributed to] Christopher Morley, American author and journalist (1890-1957).

A different story: Carl Sagan, Contact, Chapter 24– “The Artist’s Signature.”

Yet another story:  The Pennsylvania lottery yesterday, November 18, 2006– mid-day 914, evening 945. For interpretations, see 9/14 (Feast of the Triumph of the Cross) and also the following “signature” (i.e., “denominator”):

The image “http://www.log24.com/log/pix06B/061119-Zeta6.jpg” cannot be displayed, because it contains errors.

Number theorists may prefer to
think of 945 as the smallest
odd abundant number
(Al-Baghdadi, ca. 980-1037).

Neither of these occurrences
 of 945 in mathematics seems
 particularly divine; perhaps there
are some other properties of
 this number that make it more
credible as a divine signature–
other, that is, than its occurrence
in a lottery just in time for
Jodie Foster’s birthday.

Monday, January 23, 2006

Monday January 23, 2006

Filed under: General,Geometry — Tags: , — m759 @ 6:00 pm

In Defense of Hilbert
(On His Birthday)


Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:

“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”

Harris adds in a footnote:

“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”

If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”

For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):

[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”

Some of Zach’s references:

[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].

[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990

From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:

“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”

[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….

[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.

[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.

From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:

“What Hilbert wants to secure is not the truth, but the consistency of the old analysis.  This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics.  In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one.  The statements become meaningless figures built up from signs.  Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess.  Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs.  One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess.  And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae.  By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game.  The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules.  Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color.  Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game.  Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge.  What is meant to be recognized is that a contradiction can never occur as an end formula of a proof.  Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration.  One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color.  In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game.  It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together.  Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”

Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:

“Hilbert’s program… received its share of criticism.  Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’  But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time?  I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous.  It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”

— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)

Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:

“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”

One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136.  The quotation in the English translation given there:

“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”

Corfield says of this quotation that he’d love to be told the original German.  He should consult the above references cited by Richard Zach.

For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: , — 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.'”

Saturday, November 12, 2005

Saturday November 12, 2005

Filed under: General — Tags: — m759 @ 8:00 pm
Seven is Heaven,
Eight is a Gate


(continued)

A Singer 7-Cycle

“… problems are the poetry of chess.
They demand from the composer
 the same virtues that characterize
all worthwhile art:
originality, invention,
harmony, conciseness,
complexity, and
splendid insincerity.”

Vladimir Nabokov

Friday, March 11, 2005

Friday March 11, 2005

Filed under: General — Tags: — m759 @ 4:28 am

To a Young Scholar

truth is truth, tautalogous and true; what beauty is, that’s the thing to know

Posted 11/16/2002 at 1:51 am by TheYoungScholar

To a young scholar:
Guqin
Go
Calligraphy
Painting

Posted 11/16/2002 at 8:16 am by m759

For truth and beauty combined, see
The Eight, an entry of 4/4/2003,
to which the following sketch refers.

The image “http://www.log24.com/log/pix05/050311-Springer.jpg” cannot be displayed, because it contains errors.

Thursday, May 27, 2004

Thursday May 27, 2004

Filed under: General — Tags: , — m759 @ 10:10 am

Ineluctable

On the poetry of Geoffrey Hill:

"… why read him? Because of the things he writes about—war and peace and sacrifice, and the search for meaning and the truths of the heart, and for that haunting sense that, in spite of war and terror and the indifferences that make up our daily hells, there really is some grander reality, some ineluctable presence we keep touching. There remains in Hill the daunting possibility that it may actually all cohere in the end, or at least enough of it to keep us searching for more.

There is a hard edge to Hill, a strong Calvinist streak in him, and an intelligence that reminds one of Milton….."

— Paul Mariani, review in America of Geoffrey Hill's The Orchards of Syon

"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

"Time has been unfolded into space."

James O. Coplien, Bell Labs

"Pattern and symmetry are closely related."

James O. Coplien on Symmetry Breaking

"… as the critic S. L. Goldberg puts it, 'the chapter explores the Protean transformations of matter in time . . . apprehensible only in the condition of flux . . . as object . . . and Stephen himself, as subject. In the one aspect Stephen is seeking the principles of change and the underlying substance of sensory experience; in the other, he is seeking his self among its temporal manifestations'….

— Goldberg, S.L. 'Homer and the Nightmare of History.' Modern Critical Views: James Joyce. Ed. Harold Bloom. New York: Chelsea House, 1986. 21-38."

from the Choate site of David M. Loeb

In summary:

 

James Joyce
Joyce

Aleph,
alpha:
nought,
nought,
one
:

See also Time Fold.

(By the way, Jorn Barger seems
to have emerged from seclusion.)

 

Friday, April 9, 2004

Friday April 9, 2004

Filed under: General — Tags: — m759 @ 1:00 am

Triple Crown, Part II

(See previous entry.)

The winner is Mike Sullivan, far and away.

An essay, by Sullivan's son,
from Harper's magazine, Oct. 2002 —

Horseman, Pass By:
Glory, Grief, and the Race for
the Triple Crown

by John Jeremiah Sullivan

Far back, far back in our dark soul
the horse prances.

— D. H. Lawrence  

"As opposed to the typical sportswriter, who has a passion for the subject and can put together a sentence, my father's ambition had been to Write (poetry, no less), and sports were what he knew, so he sort of stumbled onto making his living that way….

Two years ago, in May, I sat with him in his hospital room at Riverside Methodist, in Columbus….

I asked him to tell me what he remembered from all those years of writing about sports, for he had seen some things in his time…. This is what he told me:

I was at Secretariat's Derby, in '73, the year before you were born — I don't guess you were even conceived yet. That was … just beauty, you know?  He started in last place, which he tended to do. I was covering the second-place horse, which wound up being Sham. It looked like Sham's race going into the last turn, I think. The thing you have to understand is that Sham was fast, a beautiful horse. He would have had the Triple Crown in another year. And it just didn't seem like there could be anything faster than that. Everybody was watching him. It was over, more or less. And all of a sudden there was this … like, just a disruption in the corner of your eye, in your peripheral vision. And then before you could make out what it was, here Secretariat came. And then Secretariat had passed him. No one had ever seen anything run like that–a lot of the old guys said the same thing. It was like he was some other animal out there …

I wrote that down when I got back to my father's apartment, where my younger sister and I were staying the night. He lived two more months, but that was the last time I saw him alive."

Thanks to the New York Times for today's review of John Jeremiah Sullivan's new book, which includes the above.

See, too,

Words Are Events.

Wednesday, March 31, 2004

Wednesday March 31, 2004

Filed under: General — Tags: , , — m759 @ 12:25 am

To Be

A Jesuit cites Quine:

"To be is to be the value of a variable."

— Willard Van Orman Quine, cited by Joseph T. Clark, S. J., in Conventional Logic and Modern Logic: A Prelude to Transition,  Woodstock, MD: Woodstock College Press, 1952, to which Quine contributed a preface.

Quine died in 2000 on Xmas Day.

From a July 26, 2003, entry,
The Transcendent Signified,
on an essay by mathematician
Michael Harris:

Kubrick's
monolith

Harris's
slab

From a December 10, 2003, entry:

Putting Descartes Before Dehors

      

"Descartes déclare que c'est en moi, non hors de moi, en moi, non dans le monde, que je pourrais voir si quelque chose existe hors de moi."

ATRIUM, Philosophie

For further details, see ART WARS.

The above material may be regarded as commemorating the March 31 birth of René Descartes and death of H. S. M. Coxeter.

For further details, see

Plato, Pegasus, and the Evening Star.

Wednesday, December 10, 2003

Wednesday December 10, 2003

Filed under: General — Tags: — m759 @ 6:13 pm

Putting Descartes Before Dehors

      

“Descartes déclare que c’est en moi, non hors de moi, en moi, non dans le monde, que je pourrais voir si quelque chose existe hors de moi.”

ATRIUM, Philosophie

For further details, see ART WARS.

Monday, September 15, 2003

Monday September 15, 2003

Filed under: General — Tags: , — m759 @ 4:15 pm

All the King's Horses

Johnny Cash's funeral was today.

Today is also the feast day of the Protestant saint Robert Penn Warren.

Here is how Stanley Kubrick might
make a memorial stone for Cash.

"He is
the outlaw
the people
love,
the hero
dressed
in black."

Nancy
Springer,

The
Hex Witch
of Seldom

The title of this entry, "All the King's Horses," is of course a slightly altered version of the title of Robert Penn Warren's famous novel.  For the connection with horses, see my entries of

September 12, 2003, and of

September 5, 2002.

See also 

The Journey Westward and

Into the West,

as well as the beginning of Mark Helprin's novel

Winter's Tale:

"There was a white horse, on a quiet winter morning when snow covered the streets gently…." 

Friday, September 12, 2003

Friday September 12, 2003

Filed under: General — Tags: — m759 @ 3:33 pm

Into the Sunset

I just learned of Johnny’s Cash’s death.  On Google News, the headline was  Johnny Cash rides into sunset.  The source was the Bangkok Post.

“Don’t you know that
when you play at this level
there’s no ordinary venue.”

One Night in Bangkok (midi)



No Ordinary Venue

“They are the horses of a dream.
 They are not what they seem.”

The Hex Witch of Seldom, page 16

A Singer 7-Cycle
A Singer
7-Cycle

The Magnificent Seven:

CLICK HERE for 

“the adventures of filming this epic
on location in Cuernavaca, Mexico.”

“He is the outlaw the people love,
the hero dressed in black.”

The Hex Witch of Seldom,
by Nancy Springer, page 15

“Words are events.”

Walter J. Ong, Society of Jesus 

“…search for thirty-three and three…”
The Black Queen in The Eight

Friday September 12, 2003

Filed under: General — Tags: — m759 @ 3:06 pm

Commentary
on the two previous entries

On 4:04:08:

“Je ne connais que deux sortes d’êtres immuables sur la terre: les géomètres et les animaux; ils sont conduits par deux règles invariables la démonstration et l’instinct; et encore les géomètres ont-ils eu quelques disputes, mais les animaux n’ont jamais varié.”

— Voltaire, Dictionnaire Philosophique, “Des Contradictions dans les Affaires et dans les Hommes

A Singer 7-cycle

 On 4:04:08
and on
Particularity:

“El pan que se come no es pan.”

— Voltaire quoting Montesquieu
on the Pope’s declarations,
Spanish translation

Thursday, September 11, 2003

Thursday September 11, 2003

Filed under: General — Tags: , , — m759 @ 6:25 pm

Particularity

Walter J. Ong

Particularity

Upon learning of the recent death of Walter J. Ong, S. J., philosopher of language, I ordered a copy of his book

Hopkins, the Self, and God
University of Toronto Press, 1986.

As the reader of my previous entry will discover, I have a very low opinion of the literary skills of the first Christians.   This sect’s writing has, however, improved in the past two millennia.

Despite my low opinion of the early Christians, I am still not convinced their religion is totally unfounded.  Hence my ordering of the Ong book.  Since then, I have also ordered two other books, reflecting my interests in philosophical fiction (see previous entry) and in philosophy itself:

Philosophical fiction —

The Hex Witch of Seldom,
by Nancy Springer,
Penguin Putnam Inc., 2002
(See 1 Corinthians 1:26-29)

Philosophy —

Definition,
by Richard Robinson,
Fellow of Oriel College, Oxford,
Oxford U. Press, 1954, reprinted 1962.

Following the scientific advice of Niels Bohr and Freeman Dyson, I articulated on April 25, 2003, a mad theory of the mystical significance of the number 162.

Here is that theory applied to the three works named above, all three of which I received, synchronistically, today.

Page 162 of Hopkins, the Self, and God is part of the long list of references at the back of the book.  Undiscouraged by the seeming insignificance (vide my note Dogma) of this page, I looked more closely.  Behold, there was Christ…  Carol T. Christ, that is, author of The Finer Optic: The Aesthetic of Particularity in Victorian Poetry, Yale University Press, 1975. “Particularity” seemed an apt description of my “162” approach to literature, so I consulted Christ’s remarks as described in the main body of Ong’s book.

Particularity according to Christ —

“Victorian particularist aesthetics has prospered to the present time, and not only in novels.  The isolated, particularized, unique ‘good moment’ [Christ, 105], the flash of awareness at one particular instant in just the right setting, which Hopkins celebrates….”

— Ong, Hopkins, the Self, and God, p. 14

I highly recommend the rest of Ong’s remarks on particularity.

Turning to the other two of the literary trinity of books I received today….

Page 162 of The Hex Witch of Seldom has the following:

“There was a loaf of Stroehmann’s Sunbeam Bread in the grocery sack also; she and Witchie each had several slices.  Bobbi folded and compressed hers into little squares and popped each slice into her mouth all at once.”

The religious significance of this passage seems, in Ong’s Jesuit context, quite clear.

Page 162 of Definition has the following:

“Real Definition as the Search for a Key.  Mr. Santayana, in his book on The Sense of Beauty, made the following extremely large demands on real definition:

‘A definition <of beauty> that should really define must be nothing less than the exposition of the origin, place, and elements of beauty as an object of human experience.  We must learn from it, as far as possible, why, when, and how beauty appears, what conditions an object must fulfil to be beautiful, what elements of our nature make us sensible of beauty, and what the relation is between the constitution of the object and the excitement of our sensibility.  Nothing less will really define beauty or make us understand what aesthetic appreciation is.  The definition of beauty in this sense will be the task of this whole book, a task that can be only very imperfectly accomplished within its limits.’ ”

Here is a rhetorical exercise for Jesuits that James Joyce might appreciate:

Discuss Bobbi’s “little squares” of bread as the Body of Christ.  Formulate, using Santayana’s criteria, a definition of beauty that includes this sacrament.

Refer, if necessary, to
the log24.net entries
Mr. Holland’s Week and Elegance.

Refrain from using the phrase
“scandal of particularity”
unless you can use it as well as
Annie Dillard.

Wednesday, September 10, 2003

Wednesday September 10, 2003

Filed under: General — Tags: , — m759 @ 4:04 pm

4:04:08

The title refers to my entry of last April 4,

The Eight,

and to the time of this entry.

From D. H. Lawrence and the Dialogical Principle:

“Plato’s Dialogues…are queer little novels….[I]t was the greatest pity in the world, when philosophy and fiction got split.  They used to be one, right from the days of myth.  Then they went and parted, like a nagging married couple, with Aristotle and Thomas Aquinas and that beastly Kant.  So the novel went sloppy, and philosophy went abstract-dry.  The two should come together again, in the novel.”

— pp. 154-5 in D. H. Lawrence, “The Future of the Novel,” in Study of Thomas Hardy and Other Essays. Ed.  Bruce Steele.  Cambridge: Cambridge University Press,1983. 149-55.



Philosophy



Fiction

“The wild, brilliant, alert head of St. Mawr seemed to look at her out of another world… the large, brilliant eyes of that horse looked at her with demonish question…. ‘Meet him half way,’ Lewis [the groom] said.  But halfway across from our human world to that terrific equine twilight was not a small step.”    

— pp. 30, 35 in D. H. Lawrence, “St. Mawr.” 1925.  St. Mawr and Other Stories.  Ed. Brian Finney.  Cambridge: Cambridge University Press, 1983.

See also

Plato, Pegasus, and the Evening Star.

Katherine Neville’s novel The Eight, referred to in my note of April 4, is an excellent example of how not to combine philosophy with fiction.  Lest this be thought too harsh, let me say that the New Testament offers a similarly ludicrous mixture.

On the other hand, there do exist successful combinations of philosophy with fiction… For example, The Glass Bead Game, Zen and The Art of Motorcycle Maintenance, Under the Volcano, the novels of Charles Williams, and the C. S. Lewis classic That Hideous Strength.

This entry was prompted by the appearance of the god Pan in my entry on this date last year, by Hugh Grant’s comedic encounters with Pan in “Sirens,” by Lawrence’s remarks on Pan in “St. Mawr,” and by the classic film “Picnic at Hanging Rock.”

Sunday, September 7, 2003

Sunday September 7, 2003

Filed under: General — Tags: , , , — m759 @ 11:11 pm

Horse Sense

Mathematicians are familiar with the emblem of Springer Verlag, the principal publisher of higher mathematics.

Ferdinand Springer, son of Julius Springer, founder of Springer Verlag, “was a passionate chess player and published a number of books on the subject. In 1881 this personal hobby and the name Springer led the company to adopt the knight in chess (in German, Springer) as its colophon.”

Hermann Hesse on a certain sort of serenity:

“I would like to say something more to you about cheerful serenity, the serenity of the stars and of the mind…. neither frivolity nor complacency; it is supreme insight and love, affirmation of all reality, alertness on the brink of all depths and abysses; it is a virtue of saints and of knights; it is indestructible and only increases with age and nearness to death. It is the secret of beauty and the real substance of all art.”

— From The Glass Bead Game

A saint and a knight, Jeanne d’Arc, was memorably portrayed by Milla Jovovich in The Messenger.

(Jovovich seems fated to play more-than-human characters in religious epics; see The Fifth Element.)

Another Springer, related to horses and to the accusation of witchcraft faced by Jeanne d’Arc, is Nancy Springer, the author of

The Hex Witch of Seldom.

Springer has written a number of books about horses, as well as other topics.

All of the above…. especially the parts having to do with mathematics and horses… was prompted by my redrawing today of a horse-shape within mathematics.  See my entry The Eight of April 4, 2003, and the horse-figure redrawn at right below.

 



Springer
Verlag



The
Messenger



A
7-Cycle

Believers in the story theory of truth may wish to relate the gifts of Jeanne d’Arc and of the girl in The Hex Witch of Seldom to the legend of Pegasus.  See, for instance,

Plato, Pegasus, and the Evening Star.

For another connection between mathematics and horses, see Sangaku.

Wednesday, August 6, 2003

Wednesday August 6, 2003

Filed under: General — Tags: , , — m759 @ 10:23 am

Postmodern
Postmortem

“I had a lot of fun with this audacious and exasperating book. … [which] looks more than a little like Greil Marcus’s Lipstick Traces, a ‘secret history’ tracing punk rock through May 1968….”

— Michael Harris, Institut de Mathématiques de Jussieu, Université Paris 7, review of Mathematics and the Roots of Postmodern Thought, by Vladimir Tasic, Notices of the American Mathematical Society, August 2003

For some observations on the transgressive  predecessors of punk rock, see my entry Funeral March of July 26, 2003 (the last conscious day in the life of actress Marie Trintignant — see below), which contains the following:

“Sky is high and so am I,
If you’re a viper — a vi-paah.”
The Day of the Locust,
    by Nathanael West (1939)

As I noted in another another July 26 entry, the disease of postmodernism has, it seems, now infected mathematics.  For some recent outbreaks of infection in physics, see the works referred to below.

Postmodern Fields of Physics: In his book The Dreams of Reason, H. R. Pagels focuses on the science of complexity as the most outstanding new discipline emerging in recent years….”

— “The Semiotics of ‘Postmodern’ Physics,” by Hans J. Pirner, in Symbol and Physical Knowledge: The Conceptual Structure of Physics, ed. by M. Ferrari and I.-O. Stamatescu, Springer Verlag, August 2001 

For a critical look at Pagels’s work, see Midsummer Eve’s Dream.  For a less critical look, see The Marriage of Science and Mysticism.  Pagels’s book on the so-called “science of complexity” was published in June 1988.  For more recent bullshit on complexity, see

The Critical Idiom of Postmodernity and Its Contributions to an Understanding of Complexity, by Matthew Abraham, 2000,

which describes a book on complexity theory that, besides pronouncements about physics, also provides what “could very well be called a ‘postmodern ethic.’ “

The book reviewed is Paul Cilliers’s Complexity and Postmodernism: Understanding Complex Systems.

A search for related material on Cilliers yields the following:

Janis Joplin, Postmodernist

” …’all’ is ‘one,’ … the time is ‘now’ and … ‘tomorrow never happens,’ …. as Janis Joplin says, ‘it’s all the same fucking day.’

It appears that ‘time,’ … the linear, independent notion of ‘time’ that our culture embraces, is an artifact of our abstract thinking …

The problem is that ‘tomorrow never happens’ …. Aboriginal traditionalists are well aware of this topological paradox and so was Janis Joplin. Her use of the expletive in this context is therefore easy to understand … love is never having to say ‘tomorrow.’ “

Web page citing Paul Cilliers

“That’s the dumbest thing I ever heard.”

— Ryan O’Neal in “What’s Up, Doc?”

A more realistic look at postmodernism in action is provided by the following news story:

Brutal Death of an Actress Is France’s Summertime Drama

By JOHN TAGLIABUE

The actress, Marie Trintignant, died Friday [Aug. 1, 2003] in a Paris hospital, with severe head and face injuries. Her rock star companion, Bertrand Cantat, is confined to a prison hospital….

According to news reports, Ms. Trintignant and Mr. Cantat argued violently in their hotel room in Vilnius in the early hours of [Sunday] July 27 at the end of a night spent eating and drinking….

In coming months, two films starring Ms. Trintignant are scheduled to debut, including “Janis and John” by the director Samuel Benchetrit, her estranged husband and the father of two of her four children. In it, Ms. Trintignant plays Janis Joplin.

New York Times of Aug. 5, 2003

” ‘…as a matter of fact, as we discover all the time, tomorrow never happens, man. It’s all the same f…n’ day, man!’ –Janis Joplin, at live performance in Calgary on 4th July 1970 – exactly four months before her death. (apologies for censoring her exact words which can be heard on the ‘Janis Joplin in Concert’ CD)”

Janis Joplin at FamousTexans.com

All of the above fits in rather nicely with the view of science and scientists in the C. S. Lewis classic That Hideous Strength, which I strongly recommend.

For those few who both abhor postmodernism and regard the American Mathematical Society Notices

as a sort of “holy place” of Platonism, I recommend a biblical reading–

Matthew 24:15, CEV:

“Someday you will see that Horrible Thing in the holy place….”

See also Logos and Logic for more sophisticated religious remarks, by Simone Weil, whose brother, mathematician André Weil, died five years ago today.

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