A book by this title, Richard A. Brualdi’s Combinatorial Matrix Classes ,
was published by Cambridge University Press in 2006:
For some related remarks, see The Counter (March 13, 2011).
My own work deals with combinatorial properties of matrices
of 0’s and 1’s, but in the context of Galois (i.e., finite) fields,
not the real or complex fields. Despite the generality of
their titles, Combinatorial Matrix Theory and Combinatorial
Matrix Classes do not deal with Galois matrices.
The title refers to a New York Times story about
an art exhibition that opened today.
This evening’s NY Lottery numbers: 016 and 2858.
Pictures from these links:
016 (Blackboard Jungle , 1955) —
2858 (number of a Log24 post, 2007) —
This post was suggested by today's previous post, Depth,
by Plato's Diamond, and by Rebecca Newberger Goldstein's
recent fanciful fiction about Plato.
Plato, Republic , Book II, Paul Shorey translation at Perseus—
“Consider,” [382a] said I; “would a god wish to deceive, or lie, by presenting in either word or action what is only appearance?” “I don’t know,” said he. “Don’t you know,” said I, “that the veritable lie, if the expression is permissible, is a thing that all gods and men abhor?” “What do you mean?” he said. “This,” said I, “that falsehood in the most vital part of themselves, and about their most vital concerns, is something that no one willingly accepts, but it is there above all that everyone fears it.” “I don’t understand yet either.” “That is because you suspect me of some grand meaning,” [382b] I said; “but what I mean is, that deception in the soul about realities, to have been deceived and to be blindly ignorant and to have and hold the falsehood there, is what all men would least of all accept, and it is in that case that they loathe it most of all.” “Quite so,” he said.
Related material —
A meditation from the Feast of St. Francis, 2012 —
A post from Sept. 30, 2012, the reported date of death
for British children's author Helen Nicoll —
The New Criterion on the death of Hilton Kramer —
This uncredited translation of Plato is, Google Books tells us,
by "Francis MacDonald Cornfield." The name is an error,
but the error is illuminating —
From Facebook, a photo from the Feast of St. Francis, 2013:
Neantro Saavedra-Rivano, author of the 1976 paper “Finite
Geometries in the Theory of Theta Characteristics,” in Brasilia—
On the same date, art from Inception and from Diamonds Studio
in Brazil —
For the Princeton Class of 1905 —
Joyce Carol Oates Meets Emily Dickinson.
Oates —
"It is an afternoon in autumn, near dusk.
The western sky is a spider’s web of translucent gold.
I am being brought by carriage—two horses—
muted thunder of their hooves—
along narrow country roads between hilly fields
touched with the sun’s slanted rays,
to the village of Princeton, New Jersey.
The urgent pace of the horses has a dreamlike air,
like the rocking motion of the carriage;
and whoever is driving the horses
his face I cannot see, only his back—
stiff, straight, in a tight-fitting dark coat."
Dickinson —
"Because I could not stop for Death—
He kindly stopped for me—
The Carriage held but just Ourselves—
And Immortality."
Blackboard Jungle , 1955
"We are going to keep doing this
until we get it right." — June 15, 2007
"Her wall is filled with pictures,
she gets 'em one by one" — Chuck Berry
See too a more advanced geometry lesson
that also uses the diagram pictured above.
Some narrative notes in memory of a
Bowling Green State University math professor
who reportedly died at 72 on Feb. 13—
That date in this journal and Green Fields.
See also Nine is a Vine.
Those who prefer mathematics to narrative may
also prefer to read, instead of the notes above,
some material on the dead professor's specialty,
Diophantine equations. Recommended:
Mordell on Lang and Lang on Mordell as well as
Lang's article titled
"Mordell's Review, Siegel's Letter to Mordell,
Diophantine Geometry, and 20th Century Mathematics."
Some background —
"Where have all the flowers gone?"
— Rhetorical question by Pete Seeger
"It was 1964, a critical juncture in Radcliffe's history."
— Elaine DeLott Baker in an historical account
of the 1964 Rick Fields incident mentioned in
yesterday's 5:01 AM Pete Seeger post.
Baker, the young woman caught in bed with
Fields, interests me much less than another
Radcliffe student…
Thanks to HOLLIS, here is an image from the
Freshman Register of the Radcliffe College
Class of 1964 (a publication from, of course,
1960, not 1964) —
See Collinge in this journal. She didn't know me
from Adam, but the above image has been in my
memory for some time.
Since we have all changed a good deal since
1960, I don't think reproducing the image is
much of an invasion of her (current) privacy.
"Chop Wood, Carry Water" —
title of a book by Rick Fields, who was
reportedly expelled from Harvard in 1964.
"From California to the New York island" —
words of a song popular among Pete Seeger fans.
Combining these phrases, we have the following
spiritual tribute to Seeger, which may be read as
a description of his obituaries in today's news.
"Chop wood, carry water
from California
to the New York island."
See also The Dharma Bums .
(Continued from yesterday afternoon)
This journal on December 12th, 2009—
Cover illustration— Arithmetic and Music,
Borgia Apartments, The Vatican
Compare and contrast with Frenkel at the Fields Institute—
Continued from Field of Dreams, Jan. 20, 2013.
That post mentioned the March 2011 AMS Notices ,
an issue on mathematics education.
In that issue was an interview with Abel Prize winner
John Tate done in Oslo on May 25, 2010, the day
he was awarded the prize. From the interview—
|
Research Contributions
Raussen and Skau: This brings us to the next
Tate: Well, first of all, it was not a new result, except
[For a different perspective on the highlighted areas of |
"So often there is a time in mathematics for something to be done."
— John Tate in Oslo on May 25, 2010.
See also this journal on May 25, 2010, as well as
Galois Groups and Harmonic Analysis on Nov. 24, 2013.
Edward Frenkel recently claimed for Robert Langlands
the discovery of a link between two "totally different"
fields of mathematics— number theory and harmonic analysis.
He implied that before Langlands, no relationship between
these fields was known.
See his recent book, and his lecture at the Fields Institute
in Toronto on October 24, 2013.
Meanwhile, in this journal on that date, two math-related
quotations for Stephen King, author of Doctor Sleep—
|
"Danvers is a town in Essex County, Massachusetts, |
For those who prefer their mathematics presented as fact, not fiction—
(Click for a larger image.)
The arrows in the figure at the right are an attempt to say visually that
the diamond theorem is related to various fields of mathematics.
There is no claim that prior to the theorem, these fields were not related.
See also Scott Carnahan on arrow diagrams, and Mathematical Imagery.
Backstory: Frenkel's Metaphors and Waitressing for Godot.
In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.
This is false.
"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe."
— Edward Frenkel, Love and Math , 2013
For a discussion of pre-Langlands connections between
these "continents," see…
"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.
This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."
— Shanshan Ding, University of Pennsylvania graduate student
Yesterday afternoon's post linked to efforts by
the late Robert de Marrais to defend a mathematical
approach to structuralism and kaleidoscopic patterns.
Two examples of non-mathematical discourse on
such patterns:
1. A Royal Society paper from 2012—
Click the above image for related material in this journal.
2. A book by Junichi Toyota from 2009—
Kaleidoscopic Grammar: Investigation into the Nature of Binarism
I find such non-mathematical approaches much less interesting
than those based on the mathematics of reflection groups .
De Marrais described the approaches of Vladimir Arnold and,
earlier, of H. S. M. Coxeter, to such groups. These approaches
dealt only with groups of reflections in Euclidean spaces.
My own interest is in groups of reflections in Galois spaces.
See, for instance, A Simple Reflection Group of Order 168.
Galois spaces over fields of characteristic 2 are particularly
relevant to what Toyota calls binarism .
Heraclitus, Fragment 60 (Diels number):
|
The way up and the way down is one and the same. ὁδὸς ἄνω κάτω μία καὶ ὡυτή hodòs áno káto mía kaì houté |
See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.
Further context: Not Theology.
“ ‘A babbled of green fields”
— Phrase attributed to Shakespeare
quoted here on September 15th
From a New York Times piece online today,
a quote promoting science and technology,
and a quote on aptitude :
… the STEM fields (“STEM” being the current shorthand
for “science, technology, engineering and mathematics”),
which offer so much in the way of job prospects, prestige,
intellectual stimulation and income….
… scientific and mathematical aptitude at
the very highest end of the spectrum ….
From a post of June 9, 2013 :
Related material — yesterday’s posts
See as well Mood Indigo.
“ ’A babbled of green fields”
— Phrase attributed to Shakespeare
The following excerpt from a January 20, 2013, preprint shows that
a Galois-geometry version of the large Desargues 154203 configuration,
although based on the nineteenth-century work of Galois* and of Fano,**
may at times have twenty-first-century applications.
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 154203 configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428-435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106-132.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
|
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
From Andries Brouwer —
* Related material: Yesterday's evening post and The People's Cube.
(By the way, any 4×4 array is a tesseract .)
A phrase in the news recently,
"la métaphysique de l'illimité ,"
suggests a search for related material.
Found: The discussion of the metaphysics of the limitless
in Chapter Two, "The Quest: Philebus ," of Plato and the Good:
Illuminating the Darkling Vision , by Rosemary Desjardins.
See, too, the Log24 post Ayn Sof of January 7, 2011,
and A Document in Madness :
* The title is from the 2011 film version of
the 2001 novel The Dark Fields .
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103-104.
The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
I learned this afternoon of a significant death:
See a NY Times obit and "RIP, Aaron Swartz."
The latter quotes Swartz himself:
"Obviously shades of Sinclair here…"
Related material:
Not so related:
A post titled Fish Story
on secular vocabulary and San Diego.
(Content last updated 4:16 EST Jan. 12, 2013.)
Source: Rudolf Koch, The Book of Signs
The American Mathematical Society
(AMS) yesterday:
|
Lars Hörmander (1931-2012) Hörmander, who received a Fields Medal in 1962, |
Some related material:
See also posts on Damnation Morning and, from the
date of Hörmander's death,
The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."
The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.
Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .
(Naturally, Mitchell's definition of "reflection" and his statement that
"The discussion of the binary groups
applies also to the case p = 2."
should be approached with care.)
A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—
"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rn which contain the origin."
For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.
Occurrences of the phrase "magic square" in Lowe-Porter's translation of the Thomas Mann novel Doctor Faustus—
"On the wall above the piano was an arithmetical diagram fastened with drawing-pins, something he had found in a second-hand shop: a so-called magic square, such as appears also in Dürer's Melancolia , along with the hour-glass, the circle, the scale, the polyhedron, and other symbols. Here as there, the figure was divided into sixteen Arabic-numbered fields, in such a way that number one was in the right-hand lower corner, sixteen in the upper left; and the magic, or the oddity, simply consisted in the fact that the sum of these numerals, however you added them, straight down, crosswise, or diagonally, always came to thirty-four. What the principle was upon which this magic uniformity rested I never made out, but by virtue of the prominent place Adrian had given it over the piano, it always attracted the eye, and I believe I never visited his room without giving a quick glance, slanting up or straight down and testing once more the invariable, incredible result."
….
"Adrian kept without changing during the whole four and a half years he spent in Leipzig his two-room quarters in Peterstrasse near the Collegium Beatae Virginis, where he had again pinned the magic square above his cottage piano."
….
" 'The decisive factor is that every note, without exception, has significance and function according to its place in the basic series or its derivatives. That would guarantee what I call the indifference to harmony and melody.'
'A magic square,' I said. 'But do you hope to have people hear all that?' "
….
" 'Extraordinarily Dürerish. You love it. First "how will I shiver after the sun"; and then the houre-glasse of the Melancolia . Is the magic square coming too?' "
….
"Here I will remind the reader of a conversation I had with Adrian on a long-ago day, the day of his sister's wedding at Buchel, as we walked round the Cow Trough. He developed for me— under pressure of a headache— his idea of the 'strict style,' derived from the way in which, as in the lied 'O lieb Madel, wie schlecht bist du ' melody and harmony are determined by the permutation of a fundamental five-note motif, the symbolic letters h, e, a, e, e-flat. He showed me the 'magic square' of a style of technique which yet developed the extreme of variety out of identical material and in which there is no longer anything unthematic, anything that could not prove itself to be a variation of an ever constant element. This style, this technique, he said, admitted no note, not one, which did not fulfil its thematic function in the whole structure— there was no longer any free note."
Review of related material—
Last night's midnight post (disambiguation), the followup 1 AM post (ambiguation), today's noon post (ambiguity), and Dürer in this journal.
The tesseracts of the noon post are related to the Dürer magic square by a well-known adjacency property.
"… the once stable 'father's depression' has been transmuted into a shifting reality that shimmered in a multiplicity of facets."
— Haim Omer, Tel-Aviv University, on Milanese ambiguation therapy,
p. 321 in "Three Styles of Constructive Therapy,"
Constructive Therapies, Vol. 2 , pp. 319-333,
ed. by Michael F. Hoyt (Guilford Press paperback, 1998)
"A kenning… is a circumlocution
used instead of an ordinary noun
in Old Norse, Old English and
later Icelandic poetry." — Wikipedia
Note the title of Tuesday's post High White in the Dark Fields.
Related material, in memory of a composer-lyricist
who died Monday (NY Times ) or Tuesday (LA Times )—
"High white noon"
— Phrase of Don DeLillo and Josefine Lyche
"Spellbinding visuals dwarf weak characters."
— Fox News review of Snow White and the Huntsman
For some stronger characters, see Limitless , a 2011 film
based on a 2001 novel by Alan Glynn, The Dark Fields .
See also St. Andrew's Day 2011 in this journal.
"Ayant été conduit par des recherches particulières
à considérer les solutions incommensurables, je suis
parvenu à quelques résultats que je crois nouveaux."
— Évariste Galois, "Sur la Théorie des Nombres"
Soon to be a major motion picture!
From Tony Rothman's review of a 2006 book by
Siobhan Roberts—
"The most engaging aspect of the book is its
chronicle of the war between geometry and algebra,
which pits Coxeter, geometry's David, against
Nicolas Bourbaki, algebra's Goliath."
The conclusion of Rothman's review—
"There is a lesson here."
Related material: a search for Galois geometry .
A book first published in hardcover in 1974—
For more-recent discussions of religious social phenomena,
see Laurie Goodstein and David V. Mason.
Background— August 30, 2006—
In the 2006 post, the above seventh symbol 110000 was
interpreted as the I Ching hexagram with topmost and
next-to-top lines solid, not broken— Hexagram 20, View .
In a different interpretation, 110000 is the binary for the decimal
number 48— representing the I Ching's Hexagram 48, The Well .
“… Max Black, the Cornell philosopher, and
others have pointed out how ‘perhaps every science
must start with metaphor and end with algebra, and
perhaps without the metaphor there would never
have been any algebra’ ….”
– Max Black, Models and Metaphors,
Cornell U. Press, 1962, page 242, as quoted
in Dramas, Fields, and Metaphors,
by Victor Witter Turner, Cornell U. Press,
paperback, 1975, page 25
The algebra is certainly clearer than either I Ching
metaphor, but is in some respects less interesting.
For a post that combines both the above I Ching
metaphors, View and Well , see Dec. 14, 2007.
In memory of scholar Elinor Ostrom,
who died today—
See "Dark Fields" in this journal
and Peter J. Cameron's weblog today.
* Phrase from "Forbidden Planet" (1956).
See previous post.
From Labyrinth of the Line (March 2, 2011)—
"… construct the Golay code by taking the 24 points
to be the points of the projective line F23 ∪ {∞}…."
— Robert A. Wilson
A simpler projective line— a Galois geometry
model of the line F2 ∪ {∞}—
Here we may consider ∞ to be modeled*
by the third square above— the Galois window .
* Update of about 1 AM Jan. 25, 2012—
This infinity-modeling is of course a poetic conceit,
not to be taken too seriously. For a serious
discussion of points at infinity and finite fields,
see (for instance) Daniel Bump's "The Group GL(2)."
"He had come a long way to this blue lawn,
and his dream must have seemed so close
that he could hardly fail to grasp it.
He did not know that it was already behind him,
somewhere back in that vast obscurity beyond the city,
where the dark fields of the republic rolled on under the night."
— The Great Gatsby
See also St. Andrew's Day, 2011, in this journal.
From the release date of the film of Alan Glynn’s
novel The Dark Fields (now retitled “Limitless“)—
“The time is now.”
Related material—
“Why does the dog wag its tail?
Because the dog is smarter than the tail.
If the tail were smarter, it would wag the dog.”
Above: Amy Adams in “Sunshine Cleaning”
“Now, I’ll open up a line of credit for you.
You’ll be wantin’ a few toys.”
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)
Plato's Diamond
The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—
Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate…. … Both in the Republic and in the Sophist there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno that the square on the diagonal is double the original square in area." – William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985 Plato's paradigm in the Meno—
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change—
Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |
Related material: Plato's Diamond by Oslo artist Josefine Lyche.
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"Remember me to her."
— Closing words of the Algis Budrys novel Rogue Moon .
Background— Some posts in this journal related to Abel or to random thoughts from his birthday.
A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—
"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
From "Sunday Dinner" in this journal—
"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"
—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010
Some table topics—
Today's midday New York Lottery numbers were 027 and 7002.
The former suggests a Galois cube, the latter a course syllabus—
CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science
An item from that syllabus:
| Six | 22 February 2011 | DES | History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2. |
Related material: A novel, PopCo , was required reading for the course.
Discuss a different novel by the same author—
Discuss the author herself, Scarlett Thomas.
Background for the discussion—
Derrida in this journal versus Charles Williams in this journal.
Related topics from the above syllabus date—
Metaphor and Gestell and Quadrat.
Some context— Midsummer Eve's Dream.
Perfect Symmetry (Oct. 2008) and Perfect Symmetry single (Dec. 2008)—
Related science…
Heinz Pagels in Perfect Symmetry (paperback, 1985), p. xvii—
The penultimate chapter of this third part of the book—
as far as speculation is concerned— describes some
recent mathematical models for the very origin of the
universe—how the fabric of space, time and matter can
be created out of absolutely nothing. What could have more
perfect symmetry than absolute nothingness? For the first
time in history, scientists have constructed mathematical
models that account for the very creation of the universe
out of nothing.
On Grand Unified Theories (GUT's) of physics (ibid., 284)—
In spite of the fact that GUTs leave deep puzzles unsolved,
they have gone a long way toward unifying the various
quantum particles. For example, many people are disturbed
by the large numbers of gluons, quarks and leptons. Part of
the appeal of the GUT idea is that this proliferation of
quantum particles is really superficial and that all the gluons
as well at the quarks and leptons may be simply viewed as
components of a few fundamental unifying fields. Under the
GUT symmetry operation these field components transform
into one another. The reason quantum particles appear to
have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object.
Related art— Puzzle and Particles…
The Diamond 16 Puzzle (compare with Keane art above)—
—and The Standard Model of particle theory—
The fact that both the puzzle and the particles appear
within a 4×4 array is of course completely coincidental.
See also a more literary approach— "The Still Point and the Wheel"—
"Anomalies must be expected along the conceptual frontier between the temporal and the eternal."
— The Death of Adam , by Marilynne Robinson, Houghton Mifflin, 1998, essay on Marguerite de Navarre
A film by Julie Taymor, Across the Universe —  Detail of the Strawberry Fields Forever Sacred Heart — 
Julie Taymor "Shinin' like a diamond, — Album "The Dark," |
"These passages suggest that the Form is a character or set of characters
common to a number of things, i.e. the feature in reality which corresponds
to a general word. But Plato also uses language which suggests not only
that the forms exist separately (χωριστά ) from all the particulars, but also
that each form is a peculiarly accurate or good particular of its own kind,
i.e. the standard particular of the kind in question or the model (παράδειγμα )
[i.e. paradigm ] to which other particulars approximate….
… Both in the Republic and in the Sophist there is a strong suggestion
that correct thinking is following out the connexions between Forms.
The model is mathematical thinking, e.g. the proof given in the Meno
that the square on the diagonal is double the original square in area."
— William and Martha Kneale, The Development of Logic,
Oxford University Press paperback, 1985
Plato's paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change* —
Monochrome figures to
colored figures
Areas to
transformations
Continuous transformations to
non-continuous transformations
Euclidean geometry to
finite geometry
Euclidean quantities to
finite fields
Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.
* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα above, along with the commentators' specific reference to
the Meno figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.
A phrase suggested by last night's New York Times obituaries—
From Milton to Milton (click to enlarge)
The "green fields" is from Shakespeare.
The above author, Vinton Adams Dearing, died* on April 6, 2005. From this journal on that date, some babbling.
"Have your people call my people." — George Carlin
* See Dearing's page 34 —
CHAPTER V
"This is an account of the discrete groups generated by reflections…."
— Regular Polytopes , by H.S.M. Coxeter (unabridged and corrected 1973 Dover reprint of the 1963 Macmillan second edition)
"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields…. Let V be an n-dimensional vector space over a field F, and let G ≤ Gln (F) be a finite group…. An element of finite order in Gl(V ) is a reflection if its fixed point space in V is a hyperplane, called the reflecting hyperplane. There are two types of reflections: the diagonalizable reflections in Gl(V ) have a single nonidentity eigenvalue which is a root of unity; the nondiagonalizable reflections in Gl(V ) are called transvections and have determinant 1 (note that they can only occur if the characteristic of F is positive)…. A reflection group is a finite group G generated by reflections."
— Julia Hartmann and Anne V. Shepler, "Reflection Groups and Differential Forms," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955-971
"… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field. The reflections in Gl(V ) not only include diagonalizable reflections (with a single nonidentity eigenvalue), but also transvections, reflections with determinant 1 which can not be diagonalized. The transvections in Gl(V ) prevent one from developing a theory of reflection groups mirroring that for Coxeter groups or complex reflection groups."
— Julia Hartmann and Anne V. Shepler, "Jacobians of Reflection Groups," Transactions of the American Mathematical Society , Vol. 360, No. 1 (2008), pp. 123-133 (Pdf available at CiteSeer.)
See also A Simple Reflection Group of Order 168 and this morning's Savage Logic.
For Julie Taymor on Fashion's Night Out…
This morning's post had a link to a video meditation from the director of
the 1985 film "Kiss of the Spider Woman"—
This film clip is echoed by lyrics, broadcast this morning, from Taymor's new Spider-Man musical—
You can fly too high and get too close to the sun.
See how the boy falls from the sky.
This morning's post and the "At Play" film it linked to featured class conflict and Brazilian natives.
For a more down-to-earth approach to these topics, see Fox Broadcasting's new series "Running Wilde."
For Kathrin Bringmann, who has been mentioned as a possible candidate for a Fields Medal.
The four Fields medal winners were announced today at the International Congress of Mathematicians in Hyderabad, India. Bringmann was not among them.
Bringmann was, however, the winner of the 2009 SASTRA Ramanujan Prize—
See The Hindu of September 30, 2009 and this journal on that date—
For more about Bringmann's work, see an article on what has been called Ramanujan's "final problem."
For another problem with a claim to this title, see "Mathematician Untangles Legendary Problem" and search in this journal for Dyson + crank.
"Honesty's the best policy."
— Miguel de Cervantes
"Liars prosper."
— Anonymous
— Epigraphs to On Writing:
A Memoir of the Craft,
by Stephen King
|
Lavender Blue, Dilly, Dilly, Lavender Green… |
|
The cruelest month continues…
"…as Newton conceived it, the distinction between
the individualities of two particles is so marked that
it is impossible for them ever to coincide or for
either of them to alter the being of the other…."
"Waves interfere with each other because they are
interchangeable and thus not distinguishable;
two processes can coincide in space and time
but two substances cannot. Thus the wave
reveals a whole new possibility of identity…."
"The concept of a field is elusive."
— Peter Pesic, Seeing Double: Shared Identities
in Physics, Philosophy, and Literature,
Chapter 6, "The Fields of Light"
From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005— as 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
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
by Ascher Wagner, U. of Birmingham, received 27 July 1977
| Journal | Geometriae Dedicata |
| Publisher | Springer Netherlands |
| Issue | Volume 9, Number 2 / June, 1980 |
[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]
Clearly the eightfold cube is a counterexample.
Unitarian Universalist Origins: Our Historic Faith—
“In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history.”
Gravity’s Rainbow–
“For every kind of vampire, there is a kind of cross.”
Unitarian minister Richard Trudeau—
“… I called the belief that
(1) Diamonds– informative, certain truths about the world– exist
the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that
(2) The theorems of Euclidean geometry are diamonds….
As the news about non-Euclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.
Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.
Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The Non-Euclidean Revolution, p. 255
H. S. M. Coxeter, 1987, introduction to Trudeau’s book—
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…
From a 1984 book review:
"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is
— Jonathan L. Alperin,
review of books on group theory,
Bulletin (New Series) of the American
Mathematical Society 10 (1984) 121, doi:
10.1090/S0273-0979-1984-15210-8
The same example
at Wolfram.com:
Citation data from Wolfram.com:
"GL(2,p) and GL(3,3) Acting on Points"
from The Wolfram Demonstrations Project,
http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
Contributed by: Ed Pegg Jr"
As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:
"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."
Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
Spider Girl
"The 'magico-religious' tarantella
is a solo dance performed
supposedly to cure…
the delirium and contortions
attributed to the bite of a spider
at harvest (summer) time."
Moral:
Life's a dance
(and Jersey girls
are tough).
who was raised in Tenafly, New Jersey–Bull on Sacred Cows:
"Poor late nineteenth-century, poor early twentieth-century! Oh, brave new world that had such people in it: people like Charles Darwin, Karl Marx, Friedrich Nietzsche, Sigmund Freud, Albert Einstein, Werner Heisenberg, Kurt Gödel. Seven people who did more than all the machine-guns and canons of the Somme Valley or the Panzer divisions of Hitler to end the old world and to create– if not the answers– at least the questions that started off the new, each one of them killing one of the sacred cows on which Western consciousness had fed for so long…."— Apostolos Doxiadis, "Writing Incompleteness-– the Play" (pdf).
See also Mathematics and Narrative.
Something Traditional —
“German Chancellor Dr. Angela Merkel is the Charlemagne Prize laureate of 2008…. The prize will be awarded on 1 May, Ascension Day.”
Something Modern —
Previously undescribed in this journal:
| A NOTE BY THE DESIGNER
“The traditional chess set, with its naturalistic images of medieval armies, suggests a game between combatants who enjoy the winning of battles. This chess set, with its articulated images of abstract force, suggests a game between contestants who enjoy the process of thinking.
The primary principle of this design… is that the operating reality or function of each piece– both its value and how it moves– is embodied in a simple self-expressive form….  Design Copyright F. Lanier Graham 1967
— F. Lanier Graham, 1967 |
Related material: this journal on the First of May, 2008, the date of last year’s Charlemagne award.
A Wealth of
Algebraic Structure
A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
— Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.
(Published online by Cambridge University Press 19 Dec 2008.)
In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)
For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):
A new combinatorial approach to M24, by R. T. Curtis. Abstract:
“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”
(Received June 15 1974)
— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.
(Published online by Cambridge University Press 24 Oct 2008.)
* For instance:
Click for details.
“Wayne C. Booth’s lifelong
study of the art of rhetoric
illuminated the means
by which authors seduce,
cajole and lie to their readers
in the service of narrative.”
— New York Times, Oct. 11, 2005
“He ends the show with Ed Ruscha’s painting ‘The End.’ But if you consult the brochure, you’ll see that it also lists one final object up above, near the ceiling. This is the green LED exit sign that directs you out of the gallery. The sign, designed by Mark Wamble, Dawn Finley and Ben Thorne of Interloop Architecture, is, like everything else here, in the Modern’s collection. Here, of course, it is also just doing its job.”
Other Christmas Day endings —
Those of W.C. Fields– see Cafe Society (April 14, 2007)– and, this year, of Eartha Kitt:
From April 12 last year:
St. Olaf College,
Northfield, Minnesota —
From The MSCS Mess
(Dept. of Mathematics, Statistics,
and Computer Science)
November 14, 2008
Volume 37, Number 9—
Math Film Festival 2008
The MSCS Department is sponsoring the second of two film-discussion evenings this Wednesday, November 19. Come to RNS 390 at 7:00 PM to see watch [sic] two short [sic]— Whatchu Know 'bout Math and Just a Finite Simple Group of Order Two— and our feature film, Good Will Hunting. Will Hunting is a mathematical genius who's living a rough life in South Boston, while being employed at a prestigious college in Boston, he's [sic] discovered by a Fields Medal winning mathematics Professor [sic] who eventually tries to get Will to turn his life around but becomes haunted by his own professional inadequacies when compared with Will. Professor Garrett will explain the “impossible problem” and its solution after the film.
Background:
Log24 entries of Wednesday, November 19, the day "Good Will Hunting" was shown:
Damnation Morning revisited and
Mathematics and Narrative continued
|
"Of man's first disobedience,
and the fruit Of that forbidden tree, whose mortal taste Brought death into the World, and all our woe…. A red apple made the rounds, |
"Do you like apples?"
— Good Will Hunting
Serious Numbers
A Yom Kippur
Meditation
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution
Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–
"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"
(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)
What does this have to do with numbers?
Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.
Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.
For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.
First in War,
First in Peace…
| Chairman George on February 22, 1999
|
 |
Thus the |
Change for Washington:
For the details, see
yale.edu/lawweb:
\Jack M. Balkin is Knight Professor of Constitutional Law and the First Amendment at Yale Law School, and the founder and director of Yale’s Information Society Project. His books and articles range over many different fields….”
| OF THE STANDARD MODEL WITH QUANTUM GRAVITY by Ashay Dharwadker Abstract
|
* See, for instance, “The Scientific Promise of Perfect Symmetry” in The New York Times of March 20, 2007.
Part I: Random Walk
Part II: X's
3/22:
Part III: O's —
A Cartoon Graveyard
in honor of the late
Gene Persson †
Today's Garfield —
See also
Midsummer Eve's Dream:
"The meeting is closed
with the lord's‡ prayer
and refreshments are served."
† Producer of plays and musicals
including Album and
The Ruling Class
‡ Lower case in honor of
Peter O'Toole, star of
the film version of
The Ruling Class.
(This film, together with
O'Toole's My Favorite Year,
may be regarded as epitomizing
Hollywood's Jesus for Jews.)
Those who prefer
less randomness
in their religion
may consult O'Toole's
more famous film work
involving Islam,
as well as
the following structure
discussed here on
the date of Persson's death:
"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah."
(See also Hamlet’s Transformation.)
The moral of this story,
it’s simple but it’s true:
Hey, the stars might lie,
but the numbers never do.
Lucy in the Sky
with Diamonds
and Sacred Heart
PARIS — Albert Hofmann, the mystical Swiss chemist who gave the world LSD, the most powerful psychotropic substance known, died Tuesday at his hilltop home near Basel, Switzerland. He was 102.
Related material:
and
a film by Julie Taymor,
Across the Universe:
Detail of the
Strawberry Fields Forever
Sacred Heart:
A song:
Julie Taymor
“Shinin’ like a diamond,
she had tombstones
in her eyes.”— Album “The Dark,”
by Guy ClarkFor related tombstones,
see May 16-19, 2006,
and April 19, 2008.Further background:
Art Wars for
Red October.
Classical Quantum
From this morning's
New York Times:
… Dr. Wheeler set the agenda for generations of theoretical physicists, using metaphor as effectively as calculus to capture the imaginations of his students and colleagues and to pose questions that would send them, minds blazing, to the barricades to confront nature….
'He rejuvenated general relativity; he made it an experimental subject and took it away from the mathematicians,' said Freeman Dyson, a theorist at the Institute for Advanced Study….
… he [Wheeler] sailed to Copenhagen to work with Bohr, the godfather of the quantum revolution, which had shaken modern science with paradoxical statements about the nature of reality.
'You can talk about people like Buddha, Jesus, Moses, Confucius, but the thing that convinced me that such people existed were the conversations with Bohr,' Dr. Wheeler said….
… Dr. Wheeler was swept up in the Manhattan Project to build an atomic bomb. To his lasting regret, the bomb was not ready in time to change the course of the war in Europe….
Dr. Wheeler continued to do government work after the war, interrupting his research to help develop the hydrogen bomb, promote the building of fallout shelters and support the Vietnam War….
… Dr. Wheeler wondered if this quantum uncertainty somehow applied to the universe and its whole history, whether it was the key to understanding why anything exists at all.
'We are no longer satisfied with insights only into particles, or fields of force, or geometry, or even space and time,' Dr. Wheeler wrote in 1981. 'Today we demand of physics some understanding of existence itself.'
At a 90th birthday celebration in 2003, Dr. Dyson said that Dr. Wheeler was part prosaic calculator, a 'master craftsman,' who decoded nuclear fission, and part poet. 'The poetic Wheeler is a prophet,' he said, 'standing like Moses on the top of Mount Pisgah, looking out over the promised land that his people will one day inherit.'"
— Dennis Overbye, The New York Times,
Monday, April 14, 2008
"point A / In a perspective
that begins again / At B"
— Wallace Stevens,
"The Rock"
From an entry today at the weblog of Lieven Le Bruyn (U. of Antwerp):
“MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias….”
The link to Kea (Marni Dee Sheppeard (pdf) of New Zealand) and a link in her Mutual Unbias III (Feb. 13) lead to the following illustration, from a talk, “Discrete phase space based on finite fields,” by William Wootters at the Perimeter Institute in 2005:
This illustration makes clear the
close relationship of MUB’s to the
finite geometry of the 4×4 square.
“Quantum Information Theory Related to Finite Geometry,”
and a comment at The n-Category Cafe,
On Spekkens’ toy system and finite geometry.
From the world of mathematics…
“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”
— Scott Carnahan at Secret Blogging Seminar, December 14, 2007
Carnahan’s remark in context:
“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GLr over function fields…. In the beginning, he drew a diagram….
If we remove all of the explanatory text, the diagram looks like this:
I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.
Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”
Compare and contrast the above…
… to the world of Rudolf Kaehr:
The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.
Another bridge…
Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–
From the world of mathematics…
“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane
Related material:
For further details on
the “diamond theory” of
Cullinane, see
Finite Geometry of the
Square and Cube.
For further details on
the “diamond theory” of
Kaehr, see
Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:
|
“Do not let me hear Anthony Hopkins Anthony Hopkins |
On Spekkens’ toy system and finite geometry
Background–
On finite geometry:
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.
Related material:
See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)
Part I:
Overview of Unix
at pangea.stanford.edu
Last revision August 2, 2004
“The Unix operating environment is organized into three layers. The innermost level of Unix is the kernel. This is the actual operating system, a single large program that always resides in memory. Sections of the code in this program are executed on behalf of users to do needed tasks, like access files or terminals. Strictly speaking, the kernel is Unix.
The next level of the Unix environment is composed of programs, commands, and utilities. In Unix, the basic commands like copying or removing files are implemented not as part of the kernel, but as individual programs, no different really from any program you could write. What we think of as the commands and utilities of Unix are simply a set of programs that have become standardized and distributed. There are hundreds of these, plus many additional utilities in the public domain that can be installed.
The final level of the Unix environment, which stands like an umbrella over the others, is the shell. The shell processes your terminal input and starts up the programs that you request. It also allows you to manipulate the environment in which those programs will execute in a way that is transparent to the program. The program can be written to handle standard cases, and then made to handle unusual cases simply by manipulating its environment, without having to have a special version of the program.” (My italics.)
Part II:
Programs
From my paper journal
on the date
“Good Will Hunting”
was released:
|
Friday, December 5, 1997 To: The executive editor, The New York Times Re: The Front Page/His Girl Friday Match the speaker with the speech–
|
||||
| The Speaker | Frame of Reference | |||
| 1. | rosebud | A. | J. Paul Getty | The front page, N.Y. Times, Monday, 12/1/97 |
| 2. | clock | B. | Joel Silver | Page 126, The New Yorker, 3/21/94 |
| 3. | act | C. | Blanche DuBois | The Elysian Fields |
| 4. | waltz | D. | Bob Geldof | People Weekly 12/8/97 |
| 5. | temple | E. | St. Michael | Heaven’s Gate |
| 6. | watch | F. | Susanna Moore | In the Cut (pbk., Dec. ’96) p. 261 |
| 7. | line | G. | Joseph Lelyveld | Page A21, The New York Times, 12/1/97 |
| 8. | chair | H. | Kylie Minogue | Page 69, People Weekly, 12/8/97 |
| 9. | religion | I. | Carol Gilligan | The Garden of Good and Evil |
| 10. | wife | J. | John Travolta | “Michael,” the movie |
| 11. | harp | K. | Shylock | Page 40, N.Y. Review of Books, 12/4/97 |
| 12. | Oscar | L. | Stephen King | The Shining (pbk., 1997), pp. 316, 317 |
“…while the scientist sees
everything that happens
in one point of space,
the poet feels
everything that happens
in one point of time…
all forming an
instantaneous and transparent
organism of events….”
Part III:
The Bourne Shell
“The binary program of the Bourne shell or a compatible program is located at /bin/sh on most Unix systems, and is still the default shell for the root superuser on many current Unix implementations.” –Wikipedia
See also
the recent comments
of root@matrix.net in
Peter Woit’s weblog.
“Hey, Carrie-Anne,
what’s your game now….”
— The Hollies, 1967
First to Illuminate
“From the History of a Simple Group” (pdf), by Jeremy Gray:
“The American mathematician A. B. Coble [1908; 1913]* seems to have been the first to illuminate the 27 lines and 28 bitangents with the elementary theory of geometries over finite fields.
The combinatorial aspects of all this are pleasant, but the mathematics is certainly not easy.”
* [Coble 1908] A. Coble, “A configuration in finite geometry isomorphic with that of the 27 lines on a cubic surface,” Johns Hopkins University Circular 7:80-88 (1908), 736-744.
[Coble 1913] A. Coble, “An application of finite geometry to the characteristic theory of the odd and even theta functions,” Trans. Amer. Math. Soc. 14 (1913), 241-276.
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
A Multicultural Farewell
to a winner of the
Nobel Prize for Literature,
the Egyptian author of
The Seventh Heaven:
Supernatural Stories —
"Jackson has identified
the seventh symbol."
— Stargate
Other versions of
the seventh symbol —
"… Max Black, the Cornell philosopher, and others have pointed out how 'perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra' …."
— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
Philosopher Max Black,
who died on this date
in 1988
“… Max Black, the Cornell philosopher, and others have pointed out how ‘perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra’ ….”
— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
Introductions
In talks at Valencia, Spain, in May through August of 2004, Alexander Borisenko, of Kharkov National University in the Ukraine, provided a detailed introduction to the topic of today’s opening lecture at ICM 2006 in Madrid:
An Introduction to Hamilton and Perelman’s Work on the Conjectures of Poincare and Thurston (pdf, 155 pages).
For a less detailed introduction, see an ICM 2006 press release (pdf, 3 pages) on Fields Medal winner Grigory Perelman.
Related material: The previous entry, “Beginnings,” and an introduction to the second-simplest two-dimensional geometry (Balanchine’s Birthday, 2003).
“How much story do you want?”
— George Balanchine
Today’s birthday: Ken Burns
Charley Reese on the republic:
“The republic died at Appomattox, and it’s been empire ever since.”
Charley Reese on Lincoln:
“Washington and Jefferson created the republic; Lincoln destroyed it.”
In closing…
A link in memory of Donald G. Higman, dead on Feb. 13, 2006, the day after Lincoln’s birthday:
His truth is marching on.
Solemn Dance
Virgil on the Elysian Fields:
Some wrestle on the sands, and some in play And games heroic pass the hours away. Those raise the song divine, and these advance In measur'd steps to form the solemn dance.
(See also the previous two entries.)
"The cover of this issue of the Bulletin is the frontispiece to a volume of Samuel de Fermat’s 1670 edition of Bachet’s Latin translation of Diophantus’s Arithmetica. This edition includes the marginalia of the editor’s father, Pierre de Fermat. Among these notes one finds the elder Fermat’s extraordinary comment [c. 1637] in connection with the Pythagorean equation
— Barry Mazur, Gade University Professor at Harvard
Mazur's concluding remarks are as follows:
Mazur has admitted, at his website, that this conclusion was an error:
"I erroneously identified the figure on the cover as Erato, muse of erotic poetry, but it seems, rather, to be Orpheus."
"Seems"?
The inscription on the frontispiece, "Obloquitur numeris septem discrimina vocum," is from a description of the Elysian Fields in Virgil's Aeneid, Book VI:
His demum exactis, perfecto munere divae, Devenere locos laetos, & amoena vireta Fortunatorum nemorum, sedesque beatas. Largior hic campos aether & lumine vestit Purpureo; solemque suum, sua sidera norunt. Pars in gramineis exercent membra palaestris, Contendunt ludo, & fulva luctanter arena: Pars pedibus plaudunt choreas, & carmina dicunt. Necnon Threicius longa cum veste sacerdos Obloquitur numeris septem discrimina vocum: Jamque eadem digitis, jam pectine pulsat eburno.
PITT: These rites compleat, they reach the flow'ry plains, The verdant groves, where endless pleasure reigns. Here glowing AEther shoots a purple ray, And o'er the region pours a double day. From sky to sky th'unwearied splendour runs, And nobler planets roll round brighter suns. Some wrestle on the sands, and some in play And games heroic pass the hours away. Those raise the song divine, and these advance In measur'd steps to form the solemn dance. There Orpheus graceful in his long attire, In seven divisions strikes the sounding lyre; Across the chords the quivering quill he flings, Or with his flying fingers sweeps the strings. DRYDEN: These holy rites perform'd, they took their way, Where long extended plains of pleasure lay. The verdant fields with those of heav'n may vie; With AEther veiled, and a purple sky: The blissful seats of happy souls below; Stars of their own, and their own suns they know. Their airy limbs in sports they exercise, And on the green contend the wrestlers prize. Some in heroic verse divinely sing, Others in artful measures lead the ring. The Thracian bard surrounded by the rest, There stands conspicuous in his flowing vest. His flying fingers, and harmonious quill, Strike seven distinguish'd notes, and seven at once they fill.
It is perhaps not irrelevant that the late Lorraine Hunt Lieberson's next role would have been that of Orfeo in Gluck's "Orfeo ed Euridice." See today's earlier entries.
The poets among us may like to think of Mazur's own role as that of the lyre:
Binary Geometry
There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).
Examples:
A Living Church
A skeptic’s remark:
“…the mind is an amazing thing and it can create patterns and interconnections among things all day if you let it, regardless of whether they are real connections.”
— Xanga blogger “sejanus”
A reply from G. K. Chesterton
(Log24, Jan. 18, 2004):
“Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living. To know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast. He is always expecting to see some truth that he has never seen before.”
A meditation on
Sunday’s numbers —
24
The Star
of Venus
“He looked at the fading light
in the western sky and saw Mercury,
or perhaps it was Venus,
gleaming at him as the evening star.
Darkness and light,
the old man thought.
It is what every hero legend is about.
The darkness which is more than death,
the light which is love, like our friend
Venus here….”
— Roderick MacLeish, Prince Ombra
From Log24, Oct. 23, 2002:
An excerpt from
Robert A. Heinlein‘s
classic novel Glory Road —
|
“I have many names. What would you like to call me?” “Is one of them ‘Helen’?” She smiled like sunshine and I learned that she had dimples. She looked sixteen and in her first party dress. “You are very gracious. No, she’s not even a relative. That was many, many years ago.” Her face turned thoughtful. “Would you like to call me ‘Ettarre’?” “Is that one of your names?” “It is much like one of them, allowing for different spelling and accent. Or it could be ‘Esther’ just as closely. Or ‘Aster.’ Or even ‘Estrellita.’ ” ” ‘Aster,’ ” I repeated. “Star. Lucky Star!” |
Related material:
672 Astarte and
The Venerable Bede
(born in 672).
672 illustrated:
The Venerable Bede
and the Star of Venus
The 672 connection is, of course,
not a real connection
(in the sense of “sejanus” above)
but it is nevertheless
not without interest.
Postscript of 6 PM
A further note on the above
illustration of the 672 connection:
The late Buck Owens
(see previous entry for
Owens, Reba, and the
star of Venus)
once described
his TV series as
“a show of fat old men
and pretty young girls”
(today’s Washington Post).
A further note on
lottery hermeneutics:
Those who prefer to interpret
random numbers with the aid
of a dictionary
(as in Is Nothing Sacred?)
may be pleased to note that
“heehaw” occurs in Webster’s
New World Dictionary,
College Edition, 1960,
on page 672.
In today’s Washington Post,
Richard Harrington informs us that
“As a child, Owens worked cotton and
maize fields, taking the name Buck
from a well-liked mule….”
A Circle of Quiet
From the Harvard Math Table page:
“No Math table this week. We will reconvene next week on March 14 for a special Pi Day talk by Paul Bamberg.”
| Some friends of mine are in this band. They’re playing in a bar on Diversey, way down the bill, around… I said I’d be there. Great. Imaginary number? It’s a math joke. |
From the April 2006 Notices of the American Mathematical Society, a footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:
4 There is a growing literature in the area of postmodern commentaries of [sic] Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems to demonstrate the logical inconsistency of self-government. For a critical view of this and related developments, see Bricmont and Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael Harris’s review of the latter, “I know what you mean!” [9]….
[9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
[13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.
Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:
“Their [Sokal’s and Bricmont’s] philosophy of mathematics, for instance, is summarized in the sentence ‘A mathematical constant like
doesn’t change, even if the idea one has about it may change.’ ( p. 263). This claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax in Social Text, is criticized by anthropologist Joan Fujimura, in an article translated for IS*. Most of Fujimura’s article consists of an astonishingly bland account of the history of non-euclidean geometry, in which she points out that the ratio of the circumference to the diameter depends on the metric. Sokal and Bricmont know this, and Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s pi”, presumably to avoid trivial objections like Fujimura’s — wasted effort on both sides.32 If one insists on making trivial objections, one might recall that the theorem
that p is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to
More interestingly, one can ask what kind of object
as a creation rather than a discovery. But Platonism does correspond to the familiar experience that there is something about mathematics, and not just about other mathematicians, that precisely doesn’t let us get away with saying ‘évidemment’!38
32 There are many circles in Euclid, but no pi, so I can’t think of any other reason for Sokal to have written ‘Euclid’s pi,’ unless this anachronism was an intentional part of the hoax. Sokal’s full quotation was ‘the
33 This is not mere sophistry: the construction of models over number fields actually uses arguments of this kind. A careless construction of the equations defining modular curves may make it appear that pi is included in their field of scalars.
34 Unless you claim, like the present French Minister of Education [at the time of writing, i.e. 1999], that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus
35 Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
36 Compare Morris Hirsch’s remarks in BAMS April 94.
37 IS*, p. 38, footnote 26. Weinberg’s remarks are contained in his article “Sokal’s Hoax,” in the New York Review of Books, August 8, 1996.
38 Metaphors from virtual reality may help here.”
* Earlier defined by Harris as “Impostures Scientifiques (IS), a collection of articles compiled or commissioned by Baudouin Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”
What is the moral of all this French noise?
Perhaps that, in spite of the contemptible nonsense at last summer’s Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.
Despite his disdain for Platonism, exemplified in his remarks on the noteworthy connection of pi with the zeta function in the formula given above, Harris has performed a valuable service to mathematics by pointing out the excellent historical work of Catherine Goldstein. Ms. Goldstein has demonstrated that even a French nominalist can be a first-rate scholar. Her essay on circles that Harris cites in a French version is also available in English, and will repay the study of those who, like Barry Mazur and other Harvard savants, are much too careless with the facts of history. They should consult her “Stories of the Circle,” pp. 160-190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).
For the historically-challenged mathematicians of Harvard, this essay would provide a valuable supplement to the upcoming “Pi Day” talk by Bamberg.
For those who insist on limiting their attention to mathematics proper, and ignoring its history, a suitable Pi Day observance might include becoming familiar with various proofs of the formula, pictured above, that connects pi with the zeta function of 2. For a survey, see Robin Chapman, Evaluating Zeta(2) (pdf). Zeta functions in a much wider context will be discussed at next May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).
Women's History Month continues.
Audrey Terras, University of Maryland '64:
We cannot discuss the proof here as it requires some knowledge of zeta functions of curves over finite fields.
Charles Small, Harvard '64:
The moral is that the zeta function exhibits a subtle connection between the "global" (topological, characteristic 0) nature of the curve and its "local" (diophantine, characteristic p for all but finitely many "bad" primes p) behaviour. The full extent of this connection only becomes apparent in the context of varieties more general than curves….
"Some friends of mine
are in this band…."
— David Auburn, "Proof"
"Women and Mathematics
is a joint program of
the Institute for Advanced Study
and Princeton University."
— School of Mathematics,
1 Einstein Drive,
Princeton, New Jersey
Sudden View
From John O'Hara's Birthday:
"We stopped at the Trocadero and there was hardly anyone there. We had Lanson 1926. 'Drink up, sweet. You gotta go some. How I love music. Frère Jacques, Cuernavaca, ach du lieber August. All languages. A walking Berlitz. Berlitz sounds like you with that champagne, my sweet, or how you're gonna sound.'"
— John O'Hara, Hope of Heaven, Chapter 11, 1938
"And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance."
"Lps. The keys to. Given! A way a lone a last a loved a long the
PARIS,
1922-1939."
— James Joyce, conclusion of Finnegans Wake
"Using illustrative material from religion, myth, and culture, he starts with the descent of the dove on Jesus and ends with the poetic ramblings of James Joyce."
— Review of a biography of the Holy Spirit
Monica Potts in today's New York Times on Sybille Bedford:
"Though her works were not always widely popular, they inspired a deeply fervent following of committed admirers, starting with her first published work, A Sudden View, in 1953. Later retitled A Visit to Don Otavio, it was an account of her journey through Mexico."
… "I addressed him. 'Is Cuernavaca not below Mexico City?'
'It is low.'
'Then what is this?' Another summit had sprung up above a curve.
'At your orders, the Three Marias.'
'What are the Three Marias?'
'These.'
Later, I learned from Terry that they were the three peaks by the La Cima Pass which is indeed one of the highest passes in the Republic; and still later from experience, that before running down to anywhere in this country one must first run up some six or seven thousand feet. The descents are more alarming than the climbs. We hurtled towards Cuernavaca down unparapeted slopes with the speed and angle, if not the precision, of a scenic railway– cacti flashed past like telegraph poles, the sun was brilliant, the air like laughing gas, below an enchanting valley, and the lack of brakes became part of a general allegro accelerando."
— Sybille Bedford, A Sudden View, Counterpoint Press, Counterpoint edition (April 2003), page 77
"How continually, how startlingly, the landscape changed! Now the fields were full of stones: there was a row of dead trees. An abandoned plough, silhouetted against the sky, raised its arms to heaven in mute supplication; another planet, he reflected again, a strange planet where, if you looked a little further, beyond the Tres Marias, you would find every sort of landscape at once, the Cotswolds, Windermere, New Hampshire, the meadows of the Eure-et-Loire, even the grey dunes of Cheshire, even the Sahara, a planet upon which, in the twinkling of an eye, you could change climates, and, if you cared to think so, in the crossing of a highway, three civilizations; but beautiful, there was no denying its beauty, fatal or cleansing as it happened to be, the beauty of the Earthly Paradise itself."
— Malcolm Lowry, Under the Volcano, Harper Perennial Modern Classics, 1st Perennial Classics edition (May 1, 2000), page 10
Proposition
“… a new nation, conceived in liberty and dedicated to the proposition that all men are created equal”
— Speech, A. Lincoln, Nov. 19, 1863
Some are less equal than others.
Proof:
Jacques Herbrand, born on this date in 1908.
“Herbrand… worked on field theory, considering abelian extensions of algebraic number fields. In the few months on which he worked on this topic, Herbrand published ten papers. These papers simplify proofs of results by Kronecker, Heinrich Weber, Hilbert, Takagi and Artin. Herbrand also generalised some of the results by these workers in class field theory as well as proving some important new theorems of his own.” –MacTutor
See
Karen E. Fields, translator’s introduction to Elementary Forms of the Religious Life, by Emile Durkheim:
“Durkheim breathed the air of turn-of-the-century Paris, a place that fizzed with experiments in artistic representation, and a time when philosophy, science, and art existed in nothing like today’s isolation from one another.24
24 Judith Ryan provides an illuminating account of the links joining physics, psychology, philosophy, painting, and literature in The Vanishing Subject: Early Psychology and Literary Modernism, Chicago, University of Chicago Press, 1991.”
And today’s Crimson provides an illuminating account of Judith Ryan and (implicitly) forms of the religious life at Harvard.
Parallel Lines
Meet at Infinity
| From Log24, Dec. 16, 2005: |
From today's New York Times, a man who died (like Charlie Chaplin and W. C. Fields) on Christmas Day: |
From Log24, Dec. 6, 2002,
Santa Versus the Volcano:
Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.
— Lonnie Donegan
(d. Nov. 3, 2002)
The Rock Island Line's namesake depot
in Rock Island, Illinois
A Miniature
Rosetta Stone
John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:
“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”
For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,
The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3). For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).
The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3). It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.
“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
Mathematics and Metaphor
The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles. The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible. The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.
Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated —
“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182
This is false. See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.
It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —
From Peter Woit’s weblog, Feb. 10, 2005:
“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'”
In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that
“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two. And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….
On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”
Here is another statement of the Rosetta-stone metaphor, from Weil’s translator, Martin H. Krieger, in the A.M.S. Notices of November 2004, “Some of What Mathematicians Do” (pdf):
“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’ Given an ability to read one column, can you find its translation in the other columns? In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry. In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”
For greater clarity, see Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraico-geometric,” over finite fields), and arithmetic (i.e., number-theoretic).
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:
Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world. Both these branches of mathematics had been studied intensively but separately…. Than along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds. Mathematicians love to build bridges.”
Simon Singh: “The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s creations…. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time. Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur. ‘But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'”
If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.
Mathematical Style:
Mac Lane Memorial, Part Trois
(See also Part I and Part II.)
“We have seen that there are many diverse styles that lead to success in mathematics. Choose one mathematician… from the ones we studied whose ‘mathematical style’ you find most rewarding for you…. Identify the mathematician and describe his or her mathematical style.”
| — Sarah J. Greenwald, take-home exam from Introduction to Mathematics at Appalachian State U., Boone, North Carolina |
From today’s Harvard Crimson:
Ex-Math Prof Mac Lane, 95, Dies
[Saunders] Mac Lane was most famous for the ground-breaking paper he co-wrote with Samuel Eilenberg of Columbia in 1945 which introduced category theory, a framework to show how mathematical structures relate to each other. This branch of algebra has since influenced most mathematical fields and also has functions in philosophy and linguistics, but was first dismissed by many practical mathematicians as too abstract to be useful.
Gade University Professor of Mathematics Barry Mazur, a friend of the late Mac Lane, recalled that the paper had at first been rejected from a lower-caliber mathematical journal because the editor thought that it was “more devoid of content” than any other he had read.
“Saunders wrote back and said, ‘That’s the point,'” Mazur said. “And in some ways that’s the genius of it. It’s the barest, most Beckett-like vocabulary that incorporates the theory and nothing else.”
He likened it to a sparse grammar of nouns and verbs and a limited vocabulary that is presented “in such a deft way that it will help you understand any language you wish to understand and any language will fit into it.”
“In my hour of weakness,that old enemy
tries to steal my soul.
But when he comes
like a flood to surround me
My God will step in
and a standard he’ll raise.”
Mac Lane Memorial
In memory of Saunders Mac Lane, mathematician, who died Thursday, April 14, 2005:
“It was during these years [the late 1930’s] that he wrote his famous text A Survey of Modern Algebra with G. Birkhoff which was published in 1941. Kaplansky writes in [*] about this text:-
A Survey of Modern Algebra opened to American undergraduates what had until then been largely reserved for mathematicians in van der Waerden‘s Moderne Algebra, published a decade earlier. The impact of Birkhoff and Mac Lane on the content and teaching of algebra in colleges and universities was immediate and long sustained. What we recognise in undergraduate courses in algebra today took much of its start with the abstract algebra which they made both accessible and attractive.
Mac Lane is noted for introducing, with Eilenberg, category theory.
For some remarks on the place of category theory in the history of mathematics, see Log24 entries of Dec. 3, 2002.
Mr. Holland's Week,
continued
"Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday regardless of what might have changed in the interim. Medical science tells us that the body's cells replace themselves wholesale within every seven years, yet we tell ourselves that we are what we were.
The question is widened and elongated in the case of the Juilliard String Quartet."
— Bernard Holland in the New York Times,
Monday, May 20, 1996
"Robert Koff, a founding member of the Juilliard String Quartet and a concert violinist who performed on modern and Baroque instruments, died on Tuesday at his home in Lexington, Mass. He was 86….
Mr. Koff, along with the violinist Robert Mann, the violist Raphael Hillyer and the cellist Arthur Winograd, formed the Juilliard String Quartet in 1946…."
— Allan Kozinn in the New York Times,
Friday, February 25, 2005
"One listened, for example, to the dazed, hymnlike beauty of the F Major's Lento assai, and then to the acid that Beethoven sprinkles all around it. It is a wrestling match, awesome but also poignant. Schubert at the end of his life had already passed on to another level of spirit. Beethoven went back and forth between the temporal world and the world beyond right up to his dying day."
— Bernard Holland in the New York Times,
Monday, May 20, 1996
Words move, music moves
Only in time; but that which is only living
Can only die. Words, after speech, reach
Into the silence. Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
Not the stillness of the violin, while the note lasts,
Not that only, but the co-existence,
Or say that the end precedes the beginning,
And the end and the beginning were always there
Before the beginning and after the end.
And all is always now.
Related material: Elegance and the following description of Beethoven's last quartet.
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Program note by Eric Bromberger:
String Quartet in F major, Op. 135 This quartet – Beethoven's last complete composition – comes from the fall of 1826, one of the blackest moments in his life. During the previous two years, Beethoven had written three string quartets on commission from Prince Nikolas Galitzin, and another, the Quartet in C-sharp minor, Op. 131, composed between January and June 1826. Even then Beethoven was not done with the possibilities of the string quartet: he pressed on with yet another, making sketches for the Quartet in F major during the summer of 1826. At that point his world collapsed. His twenty-year-old nephew Karl, who had become Beethoven's ward after a bitter court fight with the boy's mother, attempted suicide. The composer was shattered: friends reported that he suddenly looked seventy years old. When the young man had recovered enough to travel, Beethoven took him – and the sketches for the new quartet – to the country home of Beethoven's brother Johann in Gneixendorf, a village about thirty miles west of Vienna. Here, as he nursed Karl back to health, Beethoven's own health began to fail. He would get up and compose at dawn, spend his days walking through the fields, and then resume composing in the evening. In Gneixendorf he completed the Quartet in F major in October and wrote a new finale to his earlier Quartet in B-flat major, Op. 130. These were his final works. When Beethoven return to Vienna in December, he took almost immediately to bed and died the following March. One would expect music composed under such turbulent circumstances to be anguished, but the Quartet in F major is radiant music, full of sunlight – it is as if Beethoven achieved in this quartet the peace unavailable to him in life. This is the shortest of the late quartets, and many critics have noted that while this music remains very much in Beethoven's late style, it returns to the classical proportions (and mood) of the Haydn quartets. The opening movement, significantly marked Allegretto rather than the expected Allegro, is the one most often cited as Haydnesque. It is in sonata form – though a sonata form without overt conflict – and Beethoven builds it on brief thematic fragments rather than long melodies. This is poised, relaxed music, and the finale cadence – on the falling figure that has run throughout the movement – is remarkable for its understatement. By contrast, the Vivace bristles with energy. Its outer sections rocket along on a sharply-syncopated main idea, while the vigorous trio sends the first violin sailing high above the other voices. The very ending is impressive: the music grows quiet, comes to a moment of stasis, and then Beethoven wrenches it to a stop with a sudden, stinging surprise. The slow movement – Beethoven carefully marks it Lento assai, cantante e tranquillo – is built on the first violin's heartfelt opening melody; the even slower middle section, full of halting rhythms, spans only ten measures before the return of the opening material, now elaborately decorated. The final movement has occasioned the most comment. In the manuscript, Beethoven noted two three-note mottoes at its beginning under the heading Der schwer gefasste Entschluss: "The Difficult Resolution." The first, solemnly intoned by viola and cello, asks the question: "Muss es sein?" ("Must it be?"). The violins' inverted answer, which comes at the Allegro, is set to the words "Es muss sein!" ("It must be!"). Coupled with the fact that this quartet is virtually Beethoven's last composition, these mottoes have given rise to a great deal of pretentious nonsense from certain commentators, mainly to the effect that they must represent Beethoven's last thoughts, a stirring philosophical affirmation of life's possibilities. The actual origins of this motto are a great deal less imposing, for they arose from a dispute over an unpaid bill, and as a private joke for friends Beethoven wrote a humorous canon on the dispute, the theme of which he then later adapted for this quartet movement. In any case, the mottoes furnish material for what turns out to be a powerful but essentially cheerful movement. The coda, which begins pizzicato, gradually gives way to bowed notes and a cadence on the "Es muss sein!" motto. |
Counting Crows
on the Feast of St. Luke
"In the fullness of time,
educated people will believe
there is no soul
independent of the body,
and hence no life after death."
— Francis Crick, who was awarded
a Nobel Prize on this date in 1962
"She went to the men on the ground and looked at them and then she found Inman apart from them. She sat and held him in her lap. He tried to talk, but she hushed him. He drifted in and out and dreamed a bright dream of a home. It had a coldwater spring rising out of a rock, black dirt fields, old trees. In his dream, the year seemed to be happening all at one time, all the seasons blending together. Apple trees hanging heavy with fruit but yet unaccountably blossoming, ice rimming the spring, okra plants blooming yellow and maroon, maple leaves red as October, corn crops tasseling, a stuffed chair pulled up to the glowing parlor hearth, pumpkins shining in the fields, laurels blooming on the hillsides, ditch banks full of orange jewelweed, white blossoms on dogwood, purple on redbud. Everything coming around at once. And there were white oaks, and a great number of crows, or at least the spirits of crows, dancing and singing in the upper limbs. There was something he wanted to say."
— Charles Frazier, Cold Mountain
The entry Ado of June 25, 2004 contains a link to an earlier entry, A Form, continued, of June 5, 2004. This in turn contains a link to a site by Wolfgang Wildgen which contains the following:
“Historically, we may say that the consequence of Bruno’s parallel work on cosmology and artificial memory is a new model of semantic fields which was so radical in its time that the first modern followers (although ignorant of this tradition) are the Von-Neumann automata and the neural net systems of the 1980s (cf. Wildgen 1998: 39, 237f).”
Wildgen, W. 1998. Das kosmische Gedächtnis. Kosmologie, Semiotik und Gedächtniskunst im Werke von Giordano Bruno. Frankfurt/Bern: Lang.
Gedächtniskunst:
Figure A
Neighborhood in a
Cellular Automaton
by Adam Campbell
For more of the Gedächtnis
in this Kunst, see the following
Google search on shc759:
Figure B
Note that the reference to “forerunners” in fig. B occurs in a journal entry of June 12, 2002. See also the reference to a journal entry of the following day, June 13, 2002, in last Tuesday’s Dirty Trick.
Those who have viewed Campbell’s applet (see fig. A) may appreciate the following observation of poet and Dante translator Robert Pinsky:
— Poetry, Computers, and Dante’s Inferno
For some related remarks
on the muses and epic poetry,
see a paper on Walter Benjamin:
“Here the memory (Gedächtnis) means
‘the epic faculty par excellence.’ “
(Benjamin, Der Erzähler, 1936: in
Gesammelte Schriften, 1991, II.2, 453)
From Lullus to Cognitive Semantics:
The Evolution of a Theory of Semantic Fields
by Wolfgang Wildgen and in
Another Page in the Foundation of Semiotics:
A Book Review of On the Composition of Images, Signs & Ideas, by Giordano Bruno…
by Mihai Nadin
“We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn’t merely sensational, that doesn’t get its message across in 10 seconds, that isn’t falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game.”
— Robert Hughes, speech of June 2, 2004
Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder.
For a meditation on the related 4×4 square grid as “art that holds time,” see Time Fold.
Deep Play
In the previous entry, there was a reference to Carl Kaysen, former director of the Institute for Advanced Study at Princeton and father of Susanna Kaysen, author of Girl, Interrupted.
A search for further information on Carl Kaysen led to
Mark Turner, Cognitive Dimensions of Social Science: The Way We Think About Politics, Economics, Law, and Society, Oxford University Press, 2001. For a draft of this work, click here.
Turner's book describes thought and culture in terms of what he calls "blends." It includes a meditation on
Clifford Geertz, "Deep Play: Notes on the Balinese Cockfight," in Dædalus, Journal of the American Academy of Arts and Sciences, issue entitled, "Myth, Symbol, and Culture," Winter 1972, volume 101, number 1
That Turner bases weighty ruminations of what he is pleased to call "social science" on the properties of cockfights suggests that the academic world is, in some respects, even more bizarre than the mental hospital described by Kaysen's daughter.
Still, Turner's concept of "blends" is not without interest.
Here is a blend based on a diagram of the fields in which Turner and Kaysen père labor:
"politics, economics,
law, and society" (Turner)
and "economics, sociology,
politics and law" (Kaysen).
In the previous entry we abstracted from the nature of these academic pursuits, representing them simply as sets in a Venn diagram. This led to the following religious icon, an example of a Turner "blend" —
The Jewel
in Venn's Lotus.
Here is another "blend," related both to the religious material in the previous entry and to Geertz's influential essay.
From my entry for
St. Patrick's Day, 2003:
Summa Theologica
How can you tell there's an Irishman
present at a cockfight?
He enters a duck.
How can you tell a Pole is present?
He bets on the duck.
How can you tell an Italian is present?
The duck wins.
(Source: Blanche Knott,
Truly Tasteless Jokes)
Illustration for the entries
of Oct. 27, 2003:
El Pato-lógico and a
"dream of heaven."
One Ring to Rule Them All
In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.
Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:
“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’ Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”
— Leonard Gillman: An Interview
This clears up a question I asked earlier in this journal….
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Wednesday, May 14, 2003 Common Sense On the mathematician Kolmogorov: “It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.” — Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003 That this theorem is Kolmogorov’s is news to me. See
The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses. Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case. A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15. This is cited by Gillman and Jerison in the classic Rings of Continuous Functions. There ARE some references that indicate Kolmogorov may have done some work of his own in this area. See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”). Any other references to Kolmogorov’s work in this area would be of interest. Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone. I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968. |
A response by Richard Cudney:
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“In regard to your entry, it is largely correct. The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works. So, I suppose my entry was unfair to Gelfand. You’re right, the issue of credit is a bit touchy since Gelfand was his student. In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student. At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst. In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A. That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up. This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras. In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique. That is to say, this whole vein of Gelfand’s work started in this joint paper. Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge. Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third). As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin. The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.” |
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:
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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.’ “ |
“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. |
” ‘…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.
Regime Change
at the New York Times:
With Honors
Departing New York Times executive editor
Howell Raines:
"Remember, when a great story breaks out,
go like hell."
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Returning |
Good Will's |
From the date "Good Will Hunting" was released:
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Friday, December 5, 1997 "Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday." To: The executive editor, The New York Times Re: The Front Page/His Girl Friday Match the speaker with the speech — |
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| The Speaker | Frame of Reference | |||
| 1. | rosebud | A. | J. Paul Getty | The front page, N.Y. Times, Monday, 12/1/97 |
| 2. | clock | B. | Joel Silver | Page 126, The New Yorker, 3/21/94 |
| 3. | act | C. | Blanche DuBois | The Elysian Fields |
| 4. | waltz | D. | Bob Geldof | People Weekly 12/8/97 |
| 5. | temple | E. | St. Michael | Heaven's Gate |
| 6. | watch | F. | Susanna Moore | In the Cut (pbk., Dec. '96) p. 261 |
| 7. | line | G. | Joseph Lelyveld | Page A21, The New York Times, 12/1/97 |
| 8. | chair | H. | Kylie Minogue | Page 69, People Weekly, 12/8/97 |
| 9. | religion | I. | Carol Gilligan | The Garden of Good and Evil |
| 10. | wife | J. | John Travolta | "Michael," the movie |
| 11. | harp | K. | Shylock | Page 40, N.Y. Review of Books, 12/4/97 |
| 12. | Oscar | L. | Stephen King | The Shining (pbk., 1997), pp. 316, 317 |
Postscript of June 5, 2003:
"…while the scientist sees everything that happens
in one point of space, the poet feels everything that happens
in one point of time … all forming an instantaneous
and transparent organism of events…."
STAR WARS
opened on this date in 1977.
From the web page Amande:
Le Christ et la Vierge apparurent souvent entourés d’une auréole en forme d’amande: la mandorle.
Étymologiquement, le mot amande est une altération de amandala, qui dérive lui-même du latin classique amygdala….
L’amande a… une connotation symbolique, celle du sexe féminin. Elle figure souvent la vulve. Elle est alors en analogie avec la yoni du vocabulaire de l’hindouisme, la vulve ou la matrice, représentée par une amande ou une noix coupée en deux.
Screenshot of the online
New York Times, May 25, 2003:
Ariel the Hutt and Princess Amygdala
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by Horia Cristescu and
The intersection of two geometric forms (lines, triangles, circles, etc.) represents forces that are even more intense than those generated by the simple forms. Such an interpenetration indicates a high level in the dynamic interaction of the correspondent energies. The empty spaces generated by such combinations are described as very efficient operational fields of the forces emanating from the central point of the YANTRA. That is why we can very often encounter representations of MANTRAS in such spaces. YANTRA and MANTRA are complementary aspects of SHIVA and their use together is much more efficient than the use of one alone.
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AMEN.
Harrowing
In memory of the many who have died on April 19, most notably Octavio Paz.
"There is a suggestion of Christ descending into the abyss for the harrowing of Hell. But it is the Consul whom we think of here, rather than of Christ."
— Introduction to Malcolm Lowry's classic novel Under the Volcano, by Stephen Spender
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"Hey, big Spender, spend a little time
with me." — Song lyric
For a somewhat deeper meditation on time, see Architecture of Eternity.
See also Literature of the Descent into Hell.
"Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child."
— Octavio Paz, quoted by Homero Aridjis
Amen.
Concluding Unscientific Postscripts:
"Once upon a time…" — Anonymous
"It's quarter to three…" — Sinatra
St. John von Neumann's Song
The mathematician John von Neumann, a heavy drinker and party animal, advocated a nuclear first strike on Moscow.* Confined to a wheelchair before his death, he was, some say, the inspiration for Kubrick's Dr. Strangelove. He was a Jew converted to Catholicism. His saint's day was February 8. Here is an excerpt from a book titled Abstract Harmonic Analysis**, just one of the fields illuminated by von Neumann's brilliance:
"…von Neumann showed that an intrinsic definition can be given for the mean M(f) of an almost periodic function…. Von Neumann proved the existence and properties of M(f) by completely elementary methods…."
Should W. B. Yeats wander into the Catholic Anticommunists' section of Paradise, he might encounter, as in "Sailing to Byzantium," an unexpected set of "singing-masters" there: the Platonic archetypes of the Hollywood Argyles.
The Argyles' attire is in keeping with Yeats's desire for gold in his "artifice of eternity"… In this case, gold lamé, but hey, it's Hollywood. The Argyles' lyrics will no doubt be somewhat more explicit in heaven. For instance, in "Alley Oop," the line
"He's a mean motor scooter and a bad go-getter"
will in its purer heavenly version be rendered
"He's a mean M(f)er and…"
in keeping with von Neumann's artifice of eternity described above.
This theological meditation was suggested by previous entries on Yeats, music and Catholicism (see Feb. 8, von Neumann's saint's day) and by the following recent weblog entries of a Harvard senior majoring in mathematics:
"I changed my profile picture to Oedipus last night because I felt cursed by fate…."
"It's not rational for me to believe that I am cursed, that the gods are set against me. Because I don't even believe in any gods!"
The spiritual benefits of a Harvard education are summarized by this student's new profile picture:
M(f)
*Source: Von Neumann and the Development of Game Theory
**by Harvard professor Lynn H. Loomis, Van Nostrand, 1953, p. 169.
Least Popular Christmas Present
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From the University of Chicago Press, Religion and Postmodernism Series:
The Gift of Death, |
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Russell Berrie, toy maker, dies on Christmas Day. (AP photo) |
See also my note "Last-Minute Shopping"
of December 20, 2002, and my note
"An AntiChristmas Present" of June 25, 2002.
On the bright side: Berrie joins comedians
W. C. Fields and Charlie Chaplin,
who also died on Christmas Day.
"Dying is easy; comedy is hard."
— Unknown source.
See my note on Santa's last words.
Waiting for Logos
Searching for background on the phrase "logos and logic" in yesterday's "Notes toward a Supreme Fact," I found this passage:
"…a theory of psychology based on the idea of the soul as the dialectical, self-contradictory syzygy of a) soul as anima and b) soul as animus. Jungian and archetypal psychology appear to have taken heed more or less of only one half of the whole syzygy, predominantly serving an anima cut loose from her own Other, the animus as logos and logic (whose first and most extreme phenomenological image is the killer of the anima, Bluebeard). Thus psychology tends to defend the virginal innocence of the anima and her imagination…"
— Wolfgang Giegerich, "Once More the Reality/Irreality Issue: A Reply to Hillman's Reply," website
The anima and other Jungian concepts are used to analyze Wallace Stevens in an excellent essay by Michael Bryson, "The Quest for the Fiction of an Absolute." Part of Bryson's motivation in this essay is the conflict between the trendy leftist nominalism of postmodern critics and the conservative realism of more traditional critics:
"David Jarraway, in his Stevens and the Question of Belief, writes about a Stevens figured as a proto-deconstructionist, insisting on 'Steven's insistence on dismantling the logocentric models of belief' (311) in 'An Ordinary Evening in New Haven.' In opposition to these readings comes a work like Janet McCann's Wallace Stevens Revisited: 'The Celestial Possible', in which the claim is made (speaking of the post-1940 period of Stevens' life) that 'God preoccupied him for the rest of his career.'"
Here "logocentric" is a buzz word for "Christian." Stevens, unlike the postmodernists, was not anti-Christian. He did, however, see that the old structures of belief could not be maintained indefinitely, and pondered what could be found to replace them. "Notes toward a Supreme Fiction" deals with this problem. In his essay on Stevens' "Notes," Bryson emphasizes the "negative capability" of Keats as a contemplative technique:
"The willingness to exist in a state of negative capability, to accept that sometimes what we are seeking is not that which reason can impose…."
For some related material, see Simone Weil's remarks on Electra waiting for her brother Orestes. Simone Weil's brother was one of the greatest mathematicians of the past century, André Weil.
"Electra did not seek Orestes, she waited for him…"
— Simone Weil
"…at the end, she pulls it all together brilliantly in the story of Electra and Orestes, where the importance of waiting on God rather than seeking is brought home forcefully."
— Tom Hinkle, review of Waiting for God
Compare her remarks on waiting for Orestes with the following passage from Waiting for God:
"We do not obtain the most precious gifts by going in search of them but by waiting for them. Man cannot discover them by his own powers, and if he sets out to seek for them he will find in their place counterfeits of which he will be unable to discern falsity.
The solution of a geometry problem does not in itself constitute a precious gift, but the same law applies to it because it is the image of something precious. Being a little fragment of particular truth, it is a pure image of the unique, eternal, and living Truth, the very Truth that once in a human voice declared: "I am the Truth."
Every school exercise, thought of in this way, is like a sacrament.
In every school exercise there is a special way of waiting upon truth, setting our hearts upon it, yet not allowing ourselves to go out in search of it. There is a way of giving our attention to the data of a problem in geometry without trying to find the solution…."
— Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of God"
Weil concludes the preceding essay with the following passage:
"Academic work is one of those fields containing a pearl so precious that it is worth while to sell all of our possessions, keeping nothing for ourselves, in order to be able to acquire it."
This biblical metaphor is also echoed in the work of Pascal, who combined in one person the theological talent of Simone Weil and the mathematical talent of her brother. After discussing how proofs should be written, Pascal says
"The method of not erring is sought by all the world. The logicians profess to guide to it, the geometricians alone attain it, and apart from their science, and the imitations of it, there are no true demonstrations. The whole art is included in the simple precepts that we have given; they alone are sufficient, they alone afford proofs; all other rules are useless or injurious. This I know by long experience of all kinds of books and persons.
And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."
— Blaise Pascal, The Art of Persuasion
Hope of Heaven
This title is taken from a John O’Hara novel I like very much. It seems appropriate because today is the birthday of three admirable public figures:
“No one can top Eleanor Powell – not even Fred Astaire.” — A fellow professional. Reportedly, “Astaire himself said she was better than him.”
That’s as good as it gets.
Let us hope that Powell, Hawkins, and Q are enjoying a place that Q, quoting Plato’s Phaedrus, described as follows:
“a fair resting-place, full of summer sounds and scents!”
This is a rather different, and more pleasant, approach to the Phaedrus than the one most familiar to later generations — that of Pirsig in Zen and the Art of Motorcyle Maintenance. Both approaches, however, display what Pirsig calls “Quality.”
One of my own generation’s closest approaches to Quality is found in the 25th Anniversary Report of the Harvard Class of 1964. Charles Small remarks,
“A lot of other stuff has gone down the drain since 1964, of course, besides my giving up being a mathematician and settling into my first retirement. My love-hate relationship with the language has intensified, and my despair with words as instruments of communion is often near total. I read a little, but not systematically. I’ve always been enthralled by the notion that Time is an illusion, a trick our minds play in an attempt to keep things separate, without any reality of its own. My experience suggests that this is literally true, but not the kind of truth that can be acted upon….
I’m always sad and always happy. As someone says in Diane Keaton’s film ‘Heaven,’ ‘It’s kind of a lost cause, but it’s a great experience.'”
I agree. Here are two links to some work of what is apparently this same Charles Small:
ART WARS:
Apollo and Dionysus
From the New York Times of October 9, 2002:
Daniel Deverell Perry, a Long Island architect who created the marble temple of art housing the Sterling and Francine Clark Art Institute in Williamstown, Mass., died Oct. 2 in Woodstock, N.Y…. He was 97.
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From The Birth of Tragedy, by Friedrich Nietzsche (tr. by Shaun Whiteside):
Chapter 1….
To the two gods of art, Apollo and Dionysus, we owe our recognition that in the Greek world there is a tremendous opposition, as regards both origins and aims, between the Apolline art of the sculptor and the non-visual, Dionysiac art of music.
Chapter 25….
From the foundation of all existence, the Dionysiac substratum of the world, no more can enter the consciousness of the human individual than can be overcome once more by that Apolline power of transfiguration, so that both of these artistic impulses are forced to unfold in strict proportion to one another, according to the law of eternal justice. Where the Dionysiac powers have risen as impetuously as we now experience them, Apollo, enveloped in a cloud, must also have descended to us; some future generation will behold his most luxuriant effects of beauty.
Notes:
“When it opened in 1955, overlooking 140 acres of fields and ponds, Arts News celebrated its elegant galleries as the ‘best organized and most highly functional museum erected anywhere.'”
Cal
References:
Force Field of Dreams
Metaphysics and chess in today’s New York Times Magazine:
Joss Whedon, creator of a new TV series —
“I’m a very hard-line, angry atheist” and
“I want to invade people’s dreams.”
Garry Kasparov on chess —
“When the computer sees forced lines,
it plays like God.”
Putting these quotations together, one is tempted to imagine God having a little game of chess with Whedon, along the lines suggested by C. S. Lewis:
As Lewis tells it the time had come for his “Adversary [as he was wont to speak of the God he had so earnestly sought to avoid] to make His final moves.” (C. S. Lewis, Surprised by Joy, Harcourt, Brace, and World, Inc., 1955, p. 216) Lewis called them “moves” because his life seemed like a chess match in which his pieces were spread all over the board in the most disadvantageous positions. The board was set for a checkmate….
For those who would like to imagine such a game (God vs. Whedon), the following may be helpful.
George Steiner has observed that
The common bond between chess, music, and mathematics may, finally, be the absence of language.
This quotation is apparently from
Fields of Force:
Fischer and Spassky at Reykjavik. by George Steiner, Viking hardcover, June 1974.
George Steiner as quoted in a review of his book Grammars of Creation:
“I put forward the intuition, provisional and qualified, that the ‘language-animal’ we have been since ancient Greece so designated us, is undergoing mutation.”
The phrase “language-animal” is telling. A Google search reveals that it is by no means a common phrase, and that Steiner may have taken it from Heidegger. From another review, by Roger Kimball:
In ”Grammars of Creation,” for example, he tells us that ”the classical and Judaic ideal of man as ‘language animal,’ as uniquely defined by the dignity of speech . . . came to an end in the antilanguage of the death camps.”
This use of the Holocaust not only gives the appearance of establishing one’s credentials as a person of great moral gravity; it also stymies criticism. Who wants to risk the charge of insensitivity by objecting that the Holocaust had nothing to do with the ”ideal of man as ‘language animal’ ”?
Steiner has about as clear an idea of the difference between “classical” and “Judaic” ideals of man as did Michael Dukakis. (See my notes of September 9, 2002.)
Clearly what music, mathematics, and chess have in common is that they are activities based on pure form, not on language. Steiner is correct to that extent. The Greeks had, of course, an extremely strong sense of form, and, indeed, the foremost philosopher of the West, Plato, based his teachings on the notion of Forms. Jews, on the other hand, have based their culture mainly on stories… that is, on language rather than on form. The phrase “language-animal” sounds much more Jewish than Greek. Steiner is himself rather adept at the manipulation of language (and of people by means of language), but, while admiring form-based disciplines, is not particularly adept at them.
I would argue that developing a strong sense of form — of the sort required to, as Lewis would have it, play chess with God — does not require any “mutation,” but merely learning two very powerful non-Jewish approaches to thought and life: the Forms of Plato and the “archetypes” of Jung as exemplified by the 64 hexagrams of the 3,000-year-old Chinese classic, the I Ching.
For a picture of how these 64 Forms, or Hexagrams, might function as a chessboard,
Other relevant links:
“As you read, watch for patterns. Pay special attention to imagery that is geometric…”
and
Evariste Galois and
The Rock That Changed Things
An article in the current New York Review of Books (dated Sept. 26) on Ursula K. Le Guin prompted me to search the Web this evening for information on a short story of hers I remembered liking. I found the following in the journal of mathematician Peter Berman:
I agree that the story is elegant (from a mathematician, a high compliment), so searched Berman’s pages further, finding this:
between The French Mathematician (a novel about Galois) and Harry Potter and the Sorcerer’s Stone.
My own version of the Philosopher’s Stone (the phrase used instead of “Sorcerer’s Stone” in the British editions of Harry Potter) appears in my profile picture at top left; see also the picture of Plato’s diamond figure in my main math website. The mathematics of finite (or “Galois”) fields plays a role in the underlying theory of this figure’s hidden symmetries. Since the perception of color plays a large role in the Le Guin story and since my version of Plato’s diamond is obtained by coloring Plato’s version, this particular “rock that changes things” might, I hope, inspire Berman to extend his table to include Le Guin’s tale as well.
Even the mosaic theme is appropriate, this being the holiest of the Mosaic holy days.
Dr. Berman, G’mar Chatimah Tova.
ART WARS of September 8, 2002:
Sunday in the Park with Forge
From The New York Times obituary section of Saturday, September 7, 2002:
Andrew Forge, 78, Painter
and a Former Dean at Yale, Dies
By ROBERTA SMITH
ndrew Forge, a painter, critic, teacher and former dean of painting at the Yale School of Art, died on Wednesday [Sept. 4] in New Milford, Conn. He was 78…
[As a painter] he reduced his formal vocabulary to two small, basic units: tiny dots and short, thin dashes of paint that he called sticks. He applied those elements meticulously, by the thousands and with continual adjustments of shape, color, orientation and density until they coalesced into luminous, optically unstable fields.
These fields occasionally gave hints of landscapes or figures, but were primarily concerned with their own internal mechanics, which unfolded to the patient viewer with a quiet, riveting lushness. In a New York Times review of Mr. Forge’s retrospective at the Yale Center for British Art in 1996, John Russell wrote that “the whole surface of the canvas is mysteriously alive, composing and recomposing itself as we come to terms with it.”
Above: Untitled image from Andrew Forge: Recent Paintings, April 2001, Bannister Gallery, Rhode Island College, Providence, RI
See also
An Essay on the work of Andrew Forge
by Karen Wilkin
in The New Criterion, September 1996
From that essay:
“At a recent dinner, the conversation—fueled, I admit, by liberal amounts of very good red wine—became a kind of Socratic dialogue about the practice of art criticism…. There was… general agreement that it’s easier to find the rapier phrase to puncture inadequate or pretentious work than to come up with a verbal equivalent for the wordless experience of being deeply moved by something you believe to be first rate.”
See also my journal note of March 22, 2001, The Matthias Defense, which begins with the epigraph
Bit by bit, putting it together.
Piece by piece, working out the vision night and day.
All it takes is time and perseverance
With a little luck along the way.
— Stephen Sondheim
History, Stephen said….
— To really know a subject you've got to learn a bit of its history….
We both know what memories can bring;
They bring diamonds and rust.
All sorts of structures that can be defined for finite sets have analogues for the projective geometry of finite fields….
Clearly this pattern is trying to tell us something; the question is what. As always, it pays to focus on the simplest case, since that's where everything starts.
In the beginning was the word….
— The Gospel according to Saint John
The anonymous author of John makes liberal use of allegory and double-entendre to illustrate this theme.
Born yesterday: Logician John Venn.
Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 nonoverlapping regions….
— History of Mathematics at St. Andrews
Who would not be rapt by the thought of such marvels?….
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The Triangle (TRIKONA)
