Log24

In 1946, Robert Graves published “King Jesus, an historical novel based on the theory and Graves’ own historical conjecture that Jesus was, in fact, the rightful heir to the Israelite throne… written while he was researching and developing his ideas for The White Goddess.”

In 1948, C. S. Lewis finished the first draft of The Lion, The Witch, and The Wardrobe, a novel in which one of the main characters is “the White Witch.”

In 1948, Robert Graves published The White Goddess.

In 1949, Robert Graves published Seven Days in New Crete [also titled Watch the North Wind Rise], “a novel about a social distopia in which Goddess worship is (once again?) the dominant religion.”

Lewis died on November 22, 1963, the day John F. Kennedy was killed.

Related material:
Log24, December 10, 2005

Graves died on December 7 (Pearl Harbor Day), 1985.

Related material:
Log24, December 7, 2005, and
Log24, December 11, 2005

Jesus died, some say, on April 7 in the year 30 A.D.

Related material:

Art Wars, April 7, 2003:
Geometry and Conceptual Art,

Eight is a Gate, and

Mathematics and Narrative.

Plato’s Diamond

— Motto of
Plato’s Academy

Proof 101

From a course description:

“This module aims to introduce the student to rigorous university level mathematics….
    Syllabus: The idea of and need for mathematical statements and proofs…. proof by contradiction… proof by induction…. the infinite number of primes….”

In the December Notices of the American Mathematical Society, Brian (E. B.) Davies, a professor of mathematics at King’s College London, questions the consistency of Peano Arithmetic (PA), which has the following axioms:

From BookRags.com—

Axiom 1. 0 is a number.

Axiom 2. The successor of any number is a number.

Axiom 3. If a and b are numbers and if their successors are equal, then a and b are equal.

Axiom 4. 0 is not the successor of any number.

Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.

It should be noted that the word “number” as used in the Peano axioms means “non-negative integer.”  The fifth axiom deserves special comment.  It is the first formal statement of what we now call the “induction axiom” or “the principle of mathematical induction.”

Peano’s fifth axiom particularly troubles Davies, who writes elsewhere:

I contend that our understanding of number should be placed in an historical context, and that the number system is a human invention.  Elementary arithmetic enables one to determine the number of primes less than twenty as certainly as anything we know.  On the other hand Peano arithmetic is a formal system, and its internal consistency is not provable, except within set-theoretic contexts which essentially already assume it, in which case their consistency is also not provable.  The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction, which is an abstraction from our everyday experience…. 
… Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated.  Even today many university students who have been taught the principle of induction prefer to avoid its use, because they do not feel that it is as natural or as certain as a purely algebraic or geometric proof, if they can find one.  The feelings of university students may not settle questions about what is truly fundamental, but they do give some insight into our native intuitions.

— E. B. Davies in
   “Counting in the real world,”
    March 2003 (word format),
    To appear in revised form in
    Brit. J. Phil. Sci. as
   “Some remarks on
    the foundations
    of quantum mechanics”

Exercise:

Discuss Davies’s claim that

The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction.

Cite the following passage in your discussion.

It will be clear by now that, if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate. 

… I can hardly do better than go back to the Greeks.  I will state and prove two of the famous theorems of Greek mathematics.  They are “simple” theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class.  Each is as fresh and significant as when it was discovered– two thousand years have not written a wrinkle on either of them.  Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader, however slender his mathematical equipment.

I. The first is Euclid’s proof of the existence of an infinity of prime numbers.

The prime numbers or primes are the numbers

   (A)   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … 

which cannot be resolved into smaller factors.  Thus 37 and 317 are prime.  The primes are the material out of which all numbers are built up by multiplication: thus

    666 = 2 ^. 3 ^. 3 ^. 37. 

Every number which is not prime itself is divisible by at least one prime (usually, of course, by several).   We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.

Let us suppose that it does, and that

   2, 3, 5, . . . , P
 
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number

   Q = (2 ^. 3 ^. 5 ^{. . . . .} P) + 1.

It is plain that Q is not divisible by any of

   2, 3, 5, …, P;

for it leaves the remainder 1 when divided by any one of these numbers.  But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them.   This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.  It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

— G. H. Hardy,
   A Mathematician’s Apology,
   quoted in the online guide for
   Clear and Simple as the Truth:
   Writing Classic Prose, by
   Francis-Noël Thomas
   and Mark Turner,
   Princeton University Press

In discussing Davies’s claim that the above proof is by induction, you may want to refer to Davies’s statement that

Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated

and to Hardy’s statement that the above proof is due to Euclid.

Holy Geometry

What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p. 10 in the Schilpp book below) that so impressed the young Albert Einstein?

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a schoolyear.  Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

(“Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am Anfang eines Schuljahres in die Hand bekam.  Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die– obwohl an sich keineswegs evident– doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien.  Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich.”)

— Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

From a website by Hans-Josef Küpper:

“Today it cannot be said with certainty which book is Einstein’s ‘holy geometry book.’  There are three different titles that come into question:

Theodor Spieker, 1890
Lehrbuch der ebenen Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.

Heinrich Borchert Lübsen, 1870
Ausführliches Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Selbstunterricht. Mit Rücksicht auf die Zwecke des praktischen Lebens.

Adolf Sickenberger, 1888
Leitfaden der elementaren Mathematik.

Young Albert Einstein owned all of these three books. The book by T. Spieker was given to him by Max Talmud (later: Talmey), a Jewish medic. The book by H. B. Lübsen was from the library of his uncle Jakob Einstein and the one of A. Sickenberger was from his parents.”

Küpper does not state clearly his source for the geometry-book information.

According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

What might be the modern version of a “holy geometry book”?

Click on picture for details.

Windmills
 

Upper part of above picture–

From today’s New York Times,
Seeing Mountains in
Starry Clouds of Creation.

Lower part of above picture–
Pilgrimage to Spider Rock:

“This magical place, according to Navajo Legend, was the home of Spider Woman, who gave the gift of weaving to the Dineh’ People.  Today’s Navajos trace the excellence of their finest textiles to this time of legends, when their patron, Changing Woman, met Spider Woman, the first Weaver.”

Vine Deloria Jr.,
 
Evolution, Creationism,
and Other Modern Myths:

“The continuing struggle between evolutionists and creationists, a hot political topic for the past four decades, took a new turn in the summer of 1999 when the Kansas Board of Education voted to omit the mention of evolution in its newly approved curriculum, setting off outraged cries of foul by the scientific establishment.  Don Quixotes on both sides mounted their chargers and went searching for windmills.”

    “Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, ‘Look, Buster, do you want to live?’….
    Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question.

Bordered version
of the sigil

The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar.  An X superimposed on a plus sign.  It looked permanent.”

— Fritz Leiber, “Damnation Morning“

For Vine Deloria Jr., who died at 72 on Sunday, Nov. 13, 2005:

        Things forgotten are shadows.
        The shadows will be as real
        as wind and rain and song and light,
        there in the old place.
        Spider Woman atop your rock,
        I would greet you,
        but I am going the other way.
        Only a fool would pursue a Navajo
        into the Canyon of Death.

From a
Beethoven’s Birthday entry:

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat

Related material:

Blue
(below),

Bee Season
(below),

Halloween Meditations,
Aquarius Jazz,
We Are the Key,
and
Jazz on St. Lucia’s Day.

“Y’know, I never imagined
the competition version involved
so many tricky permutations.”

— David Brin, Glory Season

Oslo Connection

Today is the birthday of Oystein Ore (1899-1968), Sterling Professor of Mathematics at Yale for 37 years, who was born and died in Oslo, Norway.  Ore is said to have coined the term “Galois connection.”  In his honor, an excerpt dealing with such connections:

Click to enlarge.

(Excerpt was added to Pattern Groups.)

New Page on Geometry

See Pattern Groups, which now has a link to an interesting Nov. 2003 preprint on A₆.

Today is the birthday of Sir Thomas L. Heath, a saint of geometry whose feast day is March 16.

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

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

Like footprints erased in the sand….

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

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.”

L’Affaire Dharwadker:

Non-computer proof of 4 color Theorem,
2000 Oct. 13-Nov. 30,
sci.math, 23 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 14,
sci.math, 8 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 15,
comp.misc, 2 posts

Steven Cullinane is a Liar,
2002 Nov. 1-Nov.16,
geometry.research, 2 posts

Four-colour proof claim,
2003 Aug. 10-Sept.1,
sci.math, 9 posts

Proof of 4 colour theorem No computer!!!,
2003 Aug. 10-Aug. 20,
alt.sci.math.combinatorics, 8 posts

Steven Cullinane is a Crank,
2005 July 5-July 21
sci.math, 70 posts

Permanence

“What we do may be small, but it has a certain character of permanence.”

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

For further details, see
Geometry of the 4×4 Square.

“There is no permanent place in the world for ugly mathematics.”

— Hardy, op. cit.

For further details, see
Four-colour proof claim.

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

For more on the horizon, being, and nothingness, see

• The Giant of Nothingness and
• Midsummer Eve’s Dream.

Four Quartets

Geometry Download

There is a new web page offering my notes on finite geometry in a very large (about 7 MB) zipped folder for downloading.  (Individual notes may be previewed without downloading the folder.)

Metamorphosis

This illustration was added yesterday
to Geometry of the 4×4 Square.

Geometry Update

Added a new section,
“How the MOG works,” to
Geometry of the 4×4 Square.

Geometry, continued

Added a long footnote on symplectic properties of the 4×4 array to “Geometry of the 4×4 Square.”

Writings for
Yom Kippur

by Borges and God:

Thirsty for knowing what God knows,
Juda Loew devoted himself to permutations
of letters and complex variations

New York State Lottery,
evening, Sept. 24, 2004:  185

and finally said the Name which is the Key…

New York State Lottery,
midday, Sept. 25, 2004:   673.

On 185:

See Wittgenstein’s Philosophical Investigations (PI), section 185, on the nature of rules.

On 673:

See the following works:

Moral of these writings, thanks to Gregory Chaitin:

“Mais quand une regle est fort composée, ce qui luy est conforme, passe pour irrégulier.”

[But when a rule is extremely complex, that which conforms to it passes for random.]

— Leibniz, Discours de métaphysique, VI, 1686

See also the previous entry, High Holy Hexagram, and Pi continued.

Translation Plane
for Rosh Hashanah

Figure A

From the website of Priv.-Doz. Dr. H. Klein, Arbeitsgruppe Geometrie, Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel — The Translation Plane of Order Nine There are exactly four projective planes of order nine, and one of these planes is a non-Desarguesian translation plane. Theorem. Up to isomorphism, there exists exactly one non-Desarguesian translation plane of order 9. This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.   As it turns out, the coordinatizing quasifield is a nearfield. Moreover the non-Desarguesian translation plane of order 9 has Lenz-Barlotti type IVa.3.

Two versions of the defining spreadset for this plane are shown in Figure A.  In the left part of Fig. A, the matrices of Dr. Klein are altered by the use of “2” instead of “-1” (since these are the same, modulo 3).  In the right part of Fig. A, the corresponding figures from my 1985 note Visualizing GL(2, p) are shown.

History of Mathematics

“… mathematicians often treat history with contempt (unsullied by any practice or even knowledge of it, of course).”

— The Rainbow of Mathematics

On the history of the relationship between orthogonality (in the Latin-square sense) and skewness (in the projective-space sense)–

See the newly updated

Orthogonal Latin Squares as Skew Lines.

Quilt Geometry

New Web Page:

Galois Geometry

Beyond Geometry

(Title of current L. A. art exhibit)

“The greatest obstacle to discovery
is not ignorance —
  it is the illusion of knowledge.”

Royal Roads

“Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.”

— Albert Einstein
   on his “holy geometry book”
   (entry of 3/14/04)  

“We’ll try to understand
  how people decide what is true.”

— Nathaniel Miller
    on his geometry course

“People make up stories
 about what they experience.
 Stories that catch on are called ‘true.’ “

— Richard J. Trudeau
    on his geometry course

Notes

On “Notes Toward a Supreme Fiction,” by Wallace Stevens:

“This third section continues its play of opposing forces, introducing in the second canto a ‘blue woman,’ arguably a goddess- or muse-figure, who stands apart from images of fecundity and sexuality….”

— Michael Bryson 

From a Beethoven’s Birthday entry:

Moulin Bleu

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat   

See, too, Blue Matrices, and
a link for Beethoven’s birthday:

Song for the
Unification of Europe
(Blue 1)

From today’s news:

PRAGUE, Czech Republic (AP) – Ushering in a bold new era, hundreds of thousands of people packed streets and city squares across Europe on Friday for festivals and fireworks marking the European Union’s historic enlargement to 25 countries from 15.

The expanded EU, which takes in a broad swath of the former Soviet bloc – a region separated for decades from the West by barbed wire and Cold War ideology – was widening to 450 million citizens at midnight (6 p.m.EDT) to create a collective superpower rivalling the United States.

“All these worlds are yours
except Europa.
Attempt no landing there.”

Small World

Added a note to 4×4 Geometry:

The 4×4 square model  lets us visualize the projective space PG(3,2) as well as the affine space AG(4,2).  For tetrahedral and circular models of PG(3,2), see the work of Burkard Polster.  The following is from an advertisement of a talk by Polster on PG(3,2).

Good Friday and
Descartes’s Easter Egg

“The use of z, y, x . . . to represent unknowns is due to René Descartes, in his La géometrie (1637)…. In a paper on Cartesian ovals, prepared before 1629, x alone occurs as unknown…. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown.”

— Florian Cajori, A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929. (Vol. 1, page 381)

This is from

http://members.aol.com/jeff570/variables.html.

Descartes’s Easter Egg is found at

EggMath: The Shape of an Egg —
Cartesian Ovals.  

An Easter Meditation
on Humpty Dumpty

The following is excerpted from a web page headed “Catholic Way.”  It is one of a series of vicious and stupid Roman Catholic attacks on Descartes.  Such attacks have been encouraged by the present Pope, who today said “may the culture of life and love render vain the logic of death.”

The culture of life and love is that of the geometry (if not the philosophy) of Descartes.  The logic of death is that of Karol Wojtyla, as was made very clear in the past century by the National Socialist Party, which had its roots in Roman Catholicism.

Voilà.

The upper part
of the above icon
is from EggMath.
For the lower part,
see Good Friday.

Geometry of the 4×4 Square:

http://log24.com/theory/geometry.html

“There is such a thing as a tesseract.”
— A Wrinkle in Time

Anschaulichkeit

In memory of John W. Seybold, who died at 88 on Sunday, March 14, 2004….

Seybold is said to have originated the application of the phrase “what you see is what you get” (WYSIWYG) to computerized typesetting.

The date of Seybold’s death was also the date of Einstein’s birth.

The entry “Clarity and Certainty” for that day contains a discussion by Einstein of the fact that the altitudes of a triangle have a point in common.

A March 14 search for a clear diagram of that fact yielded the above illustration, to which I returned today after reading of Seybold’s WYSIWYG philosophy.  The illustration is taken from an article by a British teacher of geometry that contains the following:

“Dick Tahta wrote… of geometry as involving the direct apprehension of imagery, gazing as into the eyes of a beloved and a certain intuition-seeing (Anschauung)…..

His sentences have tremendous power, and yet the terms he uses are slippery and seem unexplainable. What is, or what might be, ‘direct apprehension of imagery’? What is evoked by the powerfully metaphorical ‘gazing as into the eyes of a beloved’? ‘Intuition’ is a tremendously difficult term…. The combination ‘intuition-seeing’ seems to represent an attempt to convey a meaning for the German ‘Anschauung,’ and echoes the original title of the text Anschauliche Geometrie by Hilbert and Cohn-Vossen which was published in English as Geometry and the Imagination.”

From the same article:

“… for Lacan ‘mathematics … is constantly in touch with the unconscious’….

Commentators on Lacan frequently write that… he argued that the human being is captivated by an image….

The object, in a sense, gazes back.”

From a discussion group:

The final meaning above, theological contemplation, suggests that the altitude-intersection diagram above may be used for a meditation on the Trinity.  This is, of course, silly†, but no sillier than the third-rate lucubrations of the damned charlatan Lacan.

And so let us pray that Einstein on his birthday was joined by Seybold in rapturous contemplation of the Trinity as revealed‡ in the physicist’s “holy geometry book.”

† For a less silly geometrico-theological metaphor, see “Scalene Trinities” from The Mind of the Maker, by Dorothy Sayers.

‡ For a related revelation, see A Contrapuntal Theme.

Clarity and Certainty

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book* dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty [Klarheit und Sicherheit] made an indescribable impression upon me….  For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet* had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles … for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity [Sicherheit und Reinheit] in pure thinking as the Greeks showed us for the first time to be possible in geometry.”

— from “Autobiographical Notes” in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp

“Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating.”

— Carl von Clausewitz at Quotes by Clausewitz

For clarity and certainty, consult All About Altitudes (and be sure to click the “pop it up” button).

For murkiness and uncertainty, consult The Fog of War.

Happy birthday, Albert.

* Einstein’s “holy geometry booklet” was, according to Banesh Hoffman, Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

The Quality of Diamond

On February 3, 2004, archivist and abstract painter Ward Jackson died at 75.  From today’s New York Times:

“Inspired by painters like Piet Mondrian and Josef Albers, Mr. Jackson made austere, hard-edged geometric compositions, typically on diamond-shaped canvases.”

On February 3, the day that Jackson died, there were five different log24.net entries:

1. The Quality with No Name 
2. Speaking Globally
3. Lila
4. Theory of Design
5. Retiring Faculty.

Parallels with the Helguera exhibit:

Florence Foster Jenkins: Janet Jackson in (2) above.

Giulio Camillo: Myself as compiler of the synchronistic excerpts in (5).

Friedrich Froebel: David Wade in (4).

The last Shakers: Christopher Alexander and his acolytes in (1).

Ward Jackson: On Feb. 3, Jackson became a permanent part of Quality — i.e., Reality — itself, as described in (3).

Some thoughts of Hans-Georg Gadamer
relevant to Jackson’s death:

As noted in entry (3) above
on the day that Jackson died,

“All the world’s a stage.”

— William Shakespeare

Screenshot

A search on “vult decipi” at about
3:40 AM today yielded the following, from
 http://www.sacklunch.net/Latin/P/
populusvultdecipidecipiatur.html

The ad for “Geometry of Latin Squares,”
my own. is in direct competition with
“Jesus Loves You.”
Good luck, Latin squares.

For St. Emil’s Day

On this date in 1962, Emil Artin died.

He was, in his way, a priest of Apollo, god of music, light, and reason.

The previous entry dealt with permutation groups, in the context of a Jan. 2004 AMS Notices review of a book on the mathematics of juggling.

It turns out that juggling is, in fact, related to Artin’s theory of “braid groups.”  For details, see Juggling Braids.

For more on Apollo, see my entry of

1/09.

Street of the Fathers

From Bruce Wagner’s Wild Palms —

Robert Morse sings in Kyoto
as negotiators discuss
the Go chip:

“In My Room“

Coordinates for a 4×4 space:

A Small Go Board Study:

From
Université René Descartes,
45 rue des Saints Pères,
Paris

Today’s birthdays:

Kirk Douglas
Buck Henry
John Malkovich