See as well Macbeth and the Black Arts, and a report today
by The New York Times of a death on February 7th.
Monday, March 18, 2024
Dark Horse
Comments Off on Dark Horse
At the Gate
|
" 'The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness' (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable)." C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61) This is from page 157 of . . .
Flight from Eden: The Origins of |
"Time for you to see the field." — Bagger Vance.
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Saturday, March 16, 2024
For Harlan Kane: The Benjamin Interrogation
"… if the system were complete, it would turn out to have been
interrogated during the investigation of one problem or another."
Vide . . .
(Illustration updated at 6:32 AM ET Mon., March 18, 2024.)
See also the post "Fundamental Figurate Geometry"
in this journal on Monday, March 11, 2024.
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Tuesday, February 27, 2024
Macbeth and the Black Arts
“I need a photo opportunity, I want a shot at redemption.
Don’t want to end up a cartoon in a cartoon graveyard.”
— Paul Simon
Rhymin' Simon's lyrics seem particularly appropriate
in the case of the actor below, who reportedly died
on October 31 — Halloween — last year.
Earlier last October . . .
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Friday, October 6, 2023
Intersection of the Timeless with Time
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“A Shining” — New Jon Fosse Book, Out on Halloween
Shining-related material from my own life —
Other material related to my own life, also on the date
November 25, 2009, but less personal —
From the current version —
Some background —
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Tuesday, May 2, 2023
Kernel
From a post of September 24, 2011 —
Weyl on coordinate systems,
Cassirer on the kernel of being,
and A Study in Art Education.
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Tuesday, April 25, 2023
For the Crimson Abyss
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Wednesday, March 15, 2023
For Storyholics: Distilled Fire Water
". . . The last of the river diamonds . . . .
bright alluvial diamonds,
burnished clean by mountain torrents,
green and blue and yellow and red.
In the darkness, he could feel them burning,
like fire and water of the universe, distilled."
— At Play in the Fields of the Lord ,
by Peter Matthiessen (Random House, 1965)
Related Log24 posts are now tagged Fire Water.
See as well, from posts tagged Heartland Sutra —
♫ "Red and Yellow, Blue and Green"
— "Prism Song," 1964
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In Lieu of Ayahuasca
From this journal on Dec. 3, 2011 —
Some Weinberger-related art —
See as well the prose of Peter Matthiessen —
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Sunday, March 12, 2023
Monday, February 13, 2023
Monday, February 6, 2023
Interality Studies
|
You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi. |
Some remarks from this journal on April 1, 2015 —
Manifest O
|
| 83-06-21 | An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. |
The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."
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Sunday, January 22, 2023
The Stillwell Dichotomies
| Number | Space |
| Arithmetic | Geometry |
| Discrete | Continuous |
Related literature —
From a "Finite Fields in 1956" post —
The Nutshell:
Related Narrative:
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Monday, November 14, 2022
Plot Structure
From Peter J. Cameron's weblog today —
"It happens sometimes that researchers working in different fields
study the same thing, give it different names, and don’t realise that
there is further work on the subject somewhere else…."
Cameron's example of a theorem connecting work on
the same thing in different fields —
"Theorem A partition Δ is equitable for a graph Γ if and only if
the projection matrix onto the subspace of functions constant
on parts of Δ commutes with the adjacency matrix of Γ."
A phrase from Cameron's remarks today —
"Thus we have to consider 'plot structure'…."
For more remarks on different fields and plot structure , see
"Quantum Tesseract Theorem" in this weblog.
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Monday, November 7, 2022
Prescott Street Revisited: The Boys in the Kitchen
Or: MDT-48 Meets COMP360.
‘It doesn’t have a street-name and that’s because, as yet,
it doesn’t have any street profile – which is incidentally
the way we want it to stay. The boys in the kitchen are
keeping it low-key and anonymous. They’re calling it MDT-48.’
The boys in the kitchen?
— Glynn, Alan. Limitless: A Novel (p. 40).
Picador. Kindle Edition.
(Originally published by Little, Brown
in Great Britain in 2001 as The Dark Fields .)
From Log24 on Nov. 29, 2020 —
CNN story from All Souls' Day 2022 —
“This drug can be extracted from magic mushrooms,
but that is not the way our compound is generated.
It’s synthesized in a purely chemical process
to produce a crystalline form,” said Goodwin, who is
the chief medical officer of COMPASS Pathways,
the company that manufactures COMP360 and
conducted the study."
See as well "To Think That It Happened on Prescott Street"
and related posts.
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Friday, November 4, 2022
Dark Fields Lyric
Heard Thursday evening in the dark fields of the republic —
"Rebel rebel, you've torn your dress
Rebel rebel, your face is a mess
Rebel rebel, how could they know?
Hot tramp, I love you so!"
Addendum — 10:30 AM ET on 5 Nov. 2022 —
Addendum — 5:45 PM ET on 4 Nov. 2022 —
The Triangle Club of Sadness
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Tuesday, August 16, 2022
The Pristine Edge of Darkness
|
Westworld Season 4 Episode 8 (Finale)
Christina: Where am I?
Read more at: |
From a college botany laboratory in the 1915
D. H. Lawrence novel The Rainbow —
"Suddenly she had passed away into
an intensely-gleaming light of knowledge."
A later passage in the same novel, under
a metaphorical Tree of Life —
"She passed away as on a dark wind, far, far away,
into the pristine darkness of paradise, into the original
immortality. She entered the dark fields of immortality."
Some will prefer . . .
For further context, see posts tagged Screw Theory.
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Monday, August 15, 2022
The Dark Fields… Continues.
See also Dark Fields in this journal.
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Thursday, June 23, 2022
The Nutshell Suite
The above is a summary of
Pythagorean philosophy
reposted here on . . .
Battle of the Nutshells:
From a much larger nutshell
on the above Pythagorean date—
Now let's dig a bit deeper into history . . .
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Wednesday, June 22, 2022
Code Wars: “Use the Source, Luke.”
Click the above galaxy for a larger image.
"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams." — Hamlet
Battle of the Nutshells —
From a much larger nutshell
on the above code date—
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Monday, April 25, 2022
Annals of Mathematical History
Some related remarks —
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Wednesday, March 30, 2022
Games
Click to enlarge.
Related reading — George Steiner's Fields of Force , on chess in Iceland, and . . .
The New Yorker , article by Sam Knight dated March 28, 2022 —
They went to Björk’s house. She cooked salmon.
She had seen “The Witch” and introduced Eggers
to Sjón, who had written a novel about seventeenth-
century witchcraft in Iceland. When he got home,
Eggers read Sjón’s books. “I was, like, this guy’s
a fucking magician,” Eggers said. “He sees all time,
in time, out of time.”
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Friday, November 19, 2021
Fields of Consciousness
"If I'd been out 'til quarter to three
Would you lock the door,
Will you still need me, will you still feed me,
When I'm sixty-four?"
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Sunday, August 1, 2021
Freudenthal vs. Weyl
Hans Freudenthal in 1962 on the axiomatic approach to geometry
of Fano and Hilbert —
"The bond with reality is cut."
Some philosophical background —
For Weyl's "few isolated relational concepts," see (for instance)
Projective Geometries over Finite Fields , by
J. W. P. Hirschfeld (first published by Oxford University Press in 1979).
Weyl in 1932 —
|
Mathematics is the science of the infinite , its goal the symbolic comprehension of the infinite with human, that is finite, means. It is the great achievement of the Greeks to have made the contrast between the finite and the infinite fruitful for the cognition of reality. The intuitive feeling for, the quiet unquestioning acceptance of the infinite, is peculiar to the Orient; but it remains merely an abstract consciousness, which is indifferent to the concrete manifold of reality and leaves it unformed, unpenetrated. Coming from the Orient, the religious intuition of the infinite, the apeiron , takes hold of the Greek soul in the Dionysiac-Orphic epoch which precedes the Persian wars. Also in this respect the Persian wars mark the separation of the Occident from the Orient. This tension between the finite and the infinite and its conciliation now become the driving motive of Greek investigation; but every synthesis, when it has hardly been accomplished, causes the old contrast to break through anew and in a deepened sense. In this way it determines the history of theoretical cognition to our day. — "The Open World: Three Lectures on the Metaphysical Implications of Science," 1932 |
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Saturday, June 5, 2021
Bavarian Scholarship
From my search history tonight —
11:11 PM
Number Theory – BSB Catalog opacplus.bsb-muenchen.de
11:13 PM
Klein's paradox, the icosahedron, and ring class fields | SpringerLink
A resulting quotation —
"Our attempt to explain and motivate is not merely a matter of historical whimsy."
— Harvey Cohn. See also Cohn in the previous post's link to 9/11, 2014.
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Tuesday, March 23, 2021
The Cornfield Translation . . .
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Thursday, March 11, 2021
Underworld Type
" LaTeX is widely used in academia[3][4]
for the communication and
publication of scientific documents
in many fields . . . ." — Wikipedia
Related academic remarks —
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Monday, February 15, 2021
Philosophy for Emma Stone
From a post of August 30, 2015 —
“… recall the words of author Norman Mailer
that summarized his Harvard education —
‘At times, bullshit can only be countered
with superior bullshit.’ “
And at times, non-bullshit is required.
BS from The New York Times Friday on the mathematical fields
known as topology and analysis in the 1960s —
“The two fields seemed to be nearly irremediably divided,
because topology twists objects around, and analysis
needs them to be rigid.”
Some less ignorant remarks from 1986:
The above Gauss-Bonnet theorem (ca. 1848) is explained in a talk titled
“Analysis Meets Topology” labeled with the above Emma Stone date —
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Tuesday, January 5, 2021
The Shop on the Corner
George Steiner on chess —
"… the common bond between chess, music, and mathematics
may, finally, be the absence of language."
— George Steiner, Fields of Force: Fischer and Spassky at Reykjavik ,
Viking hardcover, June 1974.
In memory of George Steiner, of Walter Tevis, and of B&B Smoke Shop,
corner of Third Ave. and Liberty St., Warren, Pennsylvania, in the 1950s,
where I purchased . . .
At that point in my life, language interested me more than chess.
But I can identify with the protagonist of Walter Tevis's Queen's Gambit ,
(the book, not the film) who visited a similar smoke shop in 1960 —
… There was a long rack of magazines behind her. When she
got the cigarettes, she turned and began looking. Senator
Kennedy’s picture was on the cover of Time and Newsweek :
he was running for President . . . .
. . . Walking home with the folded [chess] magazine tucked
securely against her flat belly she thought again about that
rook move Morphy hadn’t made. The magazine said
Morphy was “perhaps the most brilliant player in the
history of the game.” The rook could come to bishop seven,
and Black had better not take it with his knight because…
She stopped, halfway down the block. A dog was barking
somewhere, and across the street from her on a well-mowed
lawn two small boys were loudly playing tag. After the
second pawn moved to king knight five, then the remaining
rook could slide over, and if the black player took
the pawn, the bishop could uncover, and if he didn’t…
She closed her eyes. If he didn’t capture it, Morphy
could force a mate in two, starting with the bishop sacrificing
itself with a check. If he did take it, the white pawn
moved again, and then the bishop went the other way
and there was nothing Black could do. There it was. One
of the little boys across the street began crying. There was
nothing Black could do. The game would be over in
twenty-nine moves at least. The way it was in the book, it
had taken Paul Morphy thirty-six moves to win. He
hadn’t seen the move with the rook. But she had.
Overhead the sun shone in a blank blue sky. The dog
continued barking. The child wailed. Beth walked slowly
home and replayed the game. Her mind was as lucid as a
perfect, stunning diamond.
***
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Wednesday, November 11, 2020
Qube
The new domain qube.link forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .
More generally, qubes.link forwards to this post,
which defines qubes .
Definition: A qube is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)
Examples: 8, 27, 64. See qubes.site.
Update on Nov. 18, 2020, at about 9:40 PM ET —
Problem:
For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and Affine Groups on Small
Binary Spaces.)
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The Work of a Comedian
Flashback to Sept. 7, 2008 —
Change for Washington:
For the details, see yale.edu/lawweb:
“As important to Chinese civilization as the Bible is to Western culture,
the I Ching or Book of Changes is one of the oldest treasures of
world literature. Yet despite many commentaries written over the years,
it is still not well understood in the English-speaking world. In this
masterful [sic ] new interpretation, Jack Balkin returns the I Ching to
its rightful place….
Jack M. Balkin
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….”
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Monday, November 9, 2020
Geometric Logic
“Strawberry Fields Forever” — Song title
“Let me take you down
’Cause I’m going to Strawberry Fields
Nothing is real
And nothing to get hung about… ” More
“Ah, but the strawberries, that’s, that’s where I had them,
they laughed at me and made jokes, but I proved
beyond the shadow of a doubt, and with geometric logic,
that a duplicate key to the wardroom icebox did exist,
and I’ve had produced that key if they hadn’t pulled
the Caine out of action. I, I know now they were only
trying to protect some fellow officer. … Naturally, I can
only cover these things from memory. If I left anything out,
why, just ask me specific questions and I’ll be glad to
answer them, one by one.” — monologuedb.com
See also The Solomon Key .
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Wednesday, November 4, 2020
Devil’s Night Game
From Devil’s Night Art Notes —
“Create the largest colour fields and win!“
The MOZAA (pronounced Mo’s-A-A ? ) game was reportedly
published on April 1, 2013. A note from this journal on that date —
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Monday, October 12, 2020
Saturday, September 19, 2020
The Summerfield Prize
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Friday, September 11, 2020
In Memoriam
|
From the Vanderbilt University obituary of Vaughan F. R. Jones —
"During the mid-1980s, while Jones was working on a problem in von Neumann algebra theory, which is related to the foundations of quantum mechanics, he discovered an unexpected link between that theory and knot theory, a mathematical field dating back to the 19th century. Specifically, he found a new mathematical expression—now known as the Jones polynomial—for distinguishing between different types of knots as well as links in three-dimensional space. Jones’ discovery had been missed by topologists during the previous 60 years, and his finding contributed to his selection as a Fields Medalist.
'Now there is an area of mathematics called said Dietmar Bisch, professor of mathematics." [Link added.] |
Related to Jones's work —
"Topological Quantum Information Theory" at
the website of Louis H. Kauffman —
http://homepages.math.uic.edu/~kauffman/Quanta.pdf.
Kauffman —
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Friday, May 1, 2020
Bullshit Studies
The following passage is from Amanda Gefter’s Trespassing
on Einstein’s Lawn (Bantam Books, 2014).
| “You know the story of Plato’s cave?” my father asked. “All the prisoners are chained up in the cave and they can’t see the real world outside, only the shadows on the wall? That’s supposed to be a negative thing, like they’ll never know reality. But the truth is, you have to be stuck inside a limited reference frame for there to be any reality at all! If you weren’t chained to your light cone, you’d see nothing. The H-state.”
I nodded. “You’d have no information. You need the broken symmetry, the shadow, to have information and information gives rise to the world. It from bit.” I couldn’t help but grin with excitement. The message was clear: having a finite frame of reference creates the illusion of a world, but even the reference frame itself is an illusion. Observers create reality, but observers aren’t real. There is nothing ontologically distinct about an observer, because you can always find a frame in which that observer disappears: the frame of the frame itself, the boundary of the boundary. “If physicists discover an invariant someday, the game will be up,” my father mused. “That would rule out the hypothesis that the universe is really nothing.” That was true. But so far, at least, every last invariant had gone the way of space and time, rendered relative and observer-dependent. Spacetime, gravity, electromagnetism, the nuclear forces, mass, energy, momentum, angular momentum, charge, dimensions, particles, fields, the vacuum, strings, the universe, the multiverse, the speed of light— one by one they had been downgraded to illusion. As the surface appearance of reality fell away, only one thing remained. Nothing. |
My path to Gefter’s father’s musing led from a quotation attributed,
probably falsely, to John Archibald Wheeler on page 52 of Octavio
Paz’s Claude Lévi-Strauss: An Introduction (Cornell, 1970) —
“There is a point at which
‘something is nothing and nothing is something.’ “
The quote may actually be by AP writer John Barbour reporting
on a 1967 American Physical Society talk by Wheeler, “The End
of Time.”
Gefter mentions Wheeler 369 times:
See as well Introduction to Quantum Woo.
Comments Off on Bullshit Studies
Thursday, March 12, 2020
Play Date
Today's 4:02 AM ET post, "Steinfeld as Rose the Hat,"
suggests a review —
A more impressive woman in white —
Update of 8 PM ET —
Beckinsale gives Oct. 5, 2001, as the date of the New York
premiere of the film "Serendipity." Synchronology check:
Beckinsale's premiere date — Oct. 5, 2001 — is incorrect.
The film was released on that date, but its New York premiere
was actually on Oct. 3, 2001. See Getty Images.
Comments Off on Play Date
Monday, March 2, 2020
What Are You?
What are you, 12?
I'm 8. What are you reading?
Just a Western.
What does that mean? Is it good?
Pretty good.
What's the story?
I haven't finished it yet.
[Link added.]
Read more: https://www.springfieldspringfield.co.uk/
movie_script.php?movie=once-upon-a-time-in-hollywood
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Thursday, February 6, 2020
Thursday, January 16, 2020
A Very Stable Cornfield
"We show deeper implications of this simple principle,
by establishing a connection with the interplay
of orbits and stabilizers of group actions."
See also Dark Fields , a post featuring a work of philosophy
translated, reportedly, by one "Francis MacDonald Cornfield" —
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Wednesday, January 1, 2020
Águila
"We learned so much about singing from each other because you get to sort of be them for a second when you're shadowing them in harmony. It's like getting on an eagle and getting to see the world through that eagle's experience." Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=linda-ronstadt-the-sound-of-my-voice
See also Aguila de Oro.
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Sunday, October 20, 2019
In Memory of Nick Tosches
See also a poem by Nick Tosches from the preceding day —
August 11, 2010 — "He Who Is of Name," in which Tosches
addresses actor James Franco (Esquire magazine).
See as well, from this journal recently . . .
Down the Rabbit Hole with James Franco —
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Friday, October 11, 2019
Dick Date (YouTube, August 7, 2013)
Down the Rabbit Hole with Stephen King —
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Sunday, September 29, 2019
Spiritual Kin
"The 15 Puzzle and the Magic Cube
are spiritual kin …."
— "Metamagical Themas" column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 20-39
As are the 15 Schoolgirls and the Eightfold Cube.
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Friday, September 27, 2019
The Black List
"… 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
Metaphor —
Algebra —
|
The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):
1.
2.
3.
4.
5.
6. SEE ALSO: Pauli Matrices REFERENCES: Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211-217, 1985. Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 80-84, 1982. Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms. New York: Plenum, pp. 47-48, 1977. Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGraw-Hill, 1964. Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980. Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955. Referenced on Wolfram|Alpha: Dirac Matrices CITE THIS AS: Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource. |
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
.
Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
Comments Off on The Black List
Monday, September 16, 2019
Emergence
"Elementary particles are the most fundamental building blocks
of nature, and their study would seem to be an expression of
simplification in its purest form. The essence of complexity
research, by contrast, is the emergence of new kinds of order
that are only manifest when systems are large and messy."
— Sean Carroll in an opinion piece that concludes as follows:
The above plug for Sean Carroll's book
The Big Picture : On the Origins of
Life, Meaning, and the Universe Itself suggests…
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Monday, September 9, 2019
The Montenegro Contingency
(For Harlan Kane)
"While digging in the grounds for the new foundation,
the broken fragments of a marble statue were unearthed."
— From Thomas Hardy, "Barbara of the House of Grebe,"
quoted in an epigraph to Paul de Man's "Shelley Disfigured,"
in turn quoted by Barbara Johnson on page 231 of Persons
and Things (Harvard paperback, 2010).
From "the world of the unintentional, the contingent, the minute,
and the particular" (Kovacevic, U. of Montenegro, 2011) —
Yes, we received your payment. No, it wasn't late, but it was for $78.13, and the bill was for $78.31. Okay, great. Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=the-circle
Another such transposition: Pages 213 and 231 in a search
for "gaps" in a 2010 paperback discussion of Lacan —
These pages are as follows —
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Wednesday, July 31, 2019
Broadway* News
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Sunday, July 14, 2019
American Pie Continues.
From Pi Day 2017 —
"Don't want to end up a cartoon in a cartoon graveyard."
“God’s plan for man in this world is Adam and Eve,
not Adam and Steve.”
— The late William E. Dannemeyer, who reportedly
died at 89 on July 9, 2019.
Hollywood offers a second opinion —
— The garden of Eden. The birthplace of Adam and Eve and Steve. — Steve? Who's Steve? — Steve is the original supermodel. The first of the purebloods.
Comments Off on American Pie Continues.
Thursday, February 28, 2019
Fooling
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of six-set geometry.
Comments Off on Fooling
Thursday, January 10, 2019
Thursday, November 1, 2018
Literary Doodles
The Great Doodle of Northrop Frye —
Shown below is a "Story Circle" based on the work of Joseph Campbell.
The author of this particular version is unknown.
Note that there are 12 steps in the above Story Circle. This suggests
some dialogue from a recent film . . .
|
Donnie —"We can't ask for help if we don't think there's anyone out there to give it. You have to grasp this concept. And that doesn't have to be fucking Jesus Christ or Buddha or Vanna White." Read more: https://www.springfieldspringfield.co.uk/movie_script.php?movie=dont-worry-he-wont-get-far-on-foot |
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Thursday, August 2, 2018
Wednesday, August 1, 2018
Fields
"Visual culture is the aspect of culture expressed in visual images.
Many academic fields study this subject, including
cultural studies, art history, critical theory, philosophy, media studies,
and anthropology." — Wikipedia
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Tuesday, July 10, 2018
A Dark and Stormy May 29th
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.
F. Scott Fitzgerald, The Great Gatsby
— Epigraph to Limitless: A Novel , by Alan Glynn
Glynn's novel was originally published in 2002 under the title
The Dark Fields .
Compare and contrast —
Stephen King's IT was first published by Viking in 1986.
See as well the May 29th date mentioned by King.
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Monday, June 11, 2018
Finite Fields in 1956
The "more complicated" material mentioned by James Propp
in the previous post was notably described by A. A. Albert in 1956:
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Sunday, June 10, 2018
Number Concept
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
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Monday, May 21, 2018
Crux
Illustration for a Warren Times Observer story of May 12, 2018 —
Related literary background —
Iacta est.
"That's the crux of it, brother."
— William Monahan's "Mojave" script
See as well a related post on
Sunset and Selma, LA.
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Friday, April 20, 2018
Friday, April 6, 2018
Watching the Zero
From "The Blacklist" Season 5, Episode 11 —
– Remind me again what it is that we think we're doing here.
– The phone acts as a passive packet sniffer.
It's a trick Tom taught me.
– Packet sniffer? Ugh.
– The FBI uses them.
I'm sure your tech people know all about them.
It can intercept and log traffic that passes over a digital network.
– It is an absolute mystery to me how these gadgets work —
the Dick Tracy phones, these blueteeth connections.
Quite frankly, I miss the rotary phone.
Except for that zero.
Watching that zero crawl back.
Oh, my God.
It was painful.
– We have the code.
– Great.
And more:
Philip J. Davis reportedly turned 86 on January 2, 2009.
An image from this journal on that date —
“You have the incorrect number.
I will tell you what you are doing:
you are turning the letter O
instead of the zero.”
— "Symbols and Signs,"
Vladimir Nabokov, 1948
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Tuesday, March 6, 2018
Fields Medal
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Wednesday, February 14, 2018
Child’s Play
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Tuesday, February 13, 2018
Babble On
Symbology for Tom Hanks and for a Latin teacher
who reportedly died on Saturday, Feb. 10, 2018 —
Click the image below to search Log24 for "green fields."
See also Space Cross.
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Monday, November 6, 2017
Babble On
A midrash on the previous post, "Spectrum at the Center" . . .
In lieu of Judy Garland singing "Somewhere Over the Spectrum,"
a flashback to a post from 2013 —
|
Babble On
|
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
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Wednesday, September 20, 2017
The Imaginary Professor
- I was a teacher. - You're being modest, aren't you? You were a professor at Boston University... Isn't that right? - Yes, well, assistant professor. - And what'd you teach? - Philosophy. Truth and logic. That sort of thing. Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=gifted
Compare and contrast with a real Boston University professor,
John Stachel, quoted here on Sept. 5, 2017.
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Friday, September 15, 2017
Space Art
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
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Thursday, August 31, 2017
A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
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Tuesday, August 22, 2017
Preacher
College to Pilot Pre-Orientation Program
for Members of
'Historically Marginalized Communities'
— Headline in The Harvard Crimson , Saturday, August 19, 2017
You know the walls along the border? They're building them with the landing pads we used in 'Nam. Think about that. —— "Preacher" Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=blood-father
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Sunday, August 6, 2017
Ides of March 2006
Recent remarks related to the July 29 death of Landon T. Clay
suggest a review of a notable figure associated with Clay.
From a 2006 obituary of mathematician George Mackey —
"A deep thinker whose work in representation theory,
group actions, and functional analysis helped
bring closer together the fields of math and physics,
Dr. Mackey died March 15 of complications from
pneumonia. He was 90, had lived in Cambridge, and
was Landon T. Clay professor emeritus at Harvard University."
— Bryan Marquard, Boston Globe, April 28, 2006
See also this journal on the date of Mackey's death (posts now tagged
Ides of March 2006).
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Friday, July 28, 2017
Aesthetic Distance
In memory of a Disney "imagineer" who reportedly died yesterday.
From the opening scene of a 2017 film, "Gifted":
Frank calls his niece Mary to breakfast on the morning she is
to enter first grade. She is dressed, for the first time, for school —
- Hey! Come on. Let's move! - No! - Let me see. - No. - Come on, I made you special breakfast. - You can't cook. - Hey, Mary, open up. (She opens her door and walks out.) - You look beautiful. - I look like a Disney character. Where's the special? - What? - You said you made me special breakfast. Read more: http://www.springfieldspringfield.co.uk/ movie_script.php?movie=gifted |
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Tuesday, June 13, 2017
Annals of Embedded Space
This journal on the above date —
Thursday, April 13, 2017
|
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Thursday, April 13, 2017
Making Space
The New York Times online today:
At MoMA, Women at Play in the Fields of Abstraction
" The famous flowchart of Modern art's evolution simply doesn't apply
in 'Making Space: Women Artists and Postwar Abstraction.' "
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Tuesday, April 4, 2017
Night at the Museum
"Do you know what he called this place? A museum."
Update of 11:06 PM ET —
A search for background on the "Holiday" screenplay leads,
via a useful historical website, to …
Other Hollywood material —
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Friday, March 24, 2017
Swimmer in the Ocean of Night
For Scarlett
From a search for "Preparation" in this journal —
"In a nutshell, the book serves as an introduction to
Gauss' theory of quadratic forms and their composition laws
(the cornerstone of his Disquisitiones Arithmeticae ) from the
modern point of view (ideals in quadratic number fields)."
From a film in which Scarlett portrays a goddess —
Madness related to several recent posts —
|
Then, with an unheard splash which sent from the silver water to the shore a line of ripples echoed in fear by my heart, a swimming thing emerged beyond the breakers. The figure may have been that of a dog, a human being, or something more strange. It could not have known that I watched—perhaps it did not care—but like a distorted fish it swam across the mirrored stars and dived beneath the surface. After a moment it came up again, and this time, since it was closer, I saw that it was carrying something across its shoulder. I knew, then, that it could be no animal, and that it was a man or something like a man, which came toward the land from a dark ocean. But it swam with a horrible ease.
— From "The Night Ocean," by H. P. Lovecraft |
As I watched, dread-filled and passive, with the fixed stare of one who awaits death in another yet knows he cannot avert it, the swimmer approached the shore—though too far down the southward beach for me to discern its outlines or features. Obscurely loping, with sparks of moonlit foam scattered by its quick gait, it emerged and was lost among the inland dunes.
Related news —
"When hard-liners seized power in Moscow in August 1991
and imprisoned Mr. Gorbachev in his vacation house on the
Black Sea, Mr. Chernyaev, a guest there and a powerful swimmer,
offered to smuggle out a note by swimming to a beach more than
three miles away. Uncertain where he could take the note, they
dropped the plan. The coup quickly failed in any case."
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Sunday, March 12, 2017
Dialogue
A Yale Law professor on Saturday, March 4, 2017 —
"Donald Trump is Shiva the Destroyer."
Related dialogue from the new film "Assassin's Creed"—
Marion Cotillard— You've thought of everything.
Jeremy Irons— Not quite. My speech. It could do
with one of your elegant openings.
Marion Cotillard— "Now I've become death,
the destroyer of worlds."
Jeremy Irons—Not sure that I could make that work.
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Tuesday, January 3, 2017
Cultist Space
The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —
The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.
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Monday, December 19, 2016
Requiem for a Pollster
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Sunday, October 2, 2016
Neville Marriner, 1924-2016
The Washington Post online today —
Neville Marriner, who led renowned
Academy of St. Martin in the Fields, dies at 92.
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Saturday, October 1, 2016
Rippling Rhythms
The previous post presented Plato's Meno diagram as
an illustration of (superimposed) yin and yang.
For those who prefer a more fluid approach to yin and yang —
From a June 15, 2016, Caltech news release on gravitational waves —
|
Audio The "chirp" tones of the two LIGO detections are available for download. Formats are suitable as ringtones for either iPhone or Android devices. (Instructions for installing custom ringtones) September 2015 Detection December 2015 Detection |
Related commentary from July 2015 and earlier —
See posts tagged Haiku.
A different perspective —
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Friday, September 9, 2016
Welcome to the Jungle*
( Sequel to the post of 12 AM Wednesday )
The following highlighted phrase was found, with a different spelling,
in The New Yorker issue dated Sept. 12, 2016.
The article in which the phrase was embedded is not recommended.
Neither is the book (which the foolhardy explorer may easily find)
from which the above snippet was taken.
* That of Fields of the Lord .
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Sunday, July 24, 2016
Point Omega …
In this post, "Omega" denotes a generic 4-element set.
For instance … Cullinane's
![Logo for 'Elements of Finite Geometry'](http://m759.net/wordpress/?s="Elements-logo.gif"){kind=logo}
or Schmeikal's
.
The mathematics appropriate for describing
group actions on such a set is not Schmeikal's
Clifford algebra, but rather Galois's finite fields.
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Friday, July 15, 2016
Autistic Enchantment*
Robert Nye, author of the novel Falstaff , reportedly died
at 77 on July 2, 2016.
Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.
In memoriam —
From the date of Nye's death:
From Nye's book:
From the date of Heinz's death:
* See also a search for the title in this journal.
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Monday, July 11, 2016
Another Manic Monday
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Friday, May 27, 2016
Peer Review
A review of the phrase "Innermost Kernel" in this journal
suggests the following meditation …
"Who am I?" — Existential cry
in "Zoolander" and "Zoolander 2."
A similar question occurs in "Peer Gynt" —
Ben Brantley in yesterday morning's print New York Times *
expressed a nihilistic view of Peer as an onion-peeler —
"Toward the end of Ibsen’s 'Peer Gynt,' a saga of self
under siege, the title character is discovered peeling
an onion, finding in the layers of that humble vegetable
a symbol for the chapters of an eventful life . . . .
… [the director’s] approach is the same one that Peer
applies to the onion: Keep stripping until you find the core.
Of course in Peer’s case what is finally found is
plenty of nothing, an apt conclusion for a man
for whom a solid self remains elusive."
I prefer a view from what Fitzgerald called
"the dark fields of the republic" — the Dordt College view —
* The Times — "A version of this review appears in print on May 26, 2016,
on page C3 of the New York edition with the headline:
'A Saga of Self-Identity, Stripped to Its Core, Still Provokes.' "
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Wednesday, April 13, 2016
Black List
A search for "Max Black" in this journal yields some images
from a post of August 30, 2006 . . .
|
"Jackson has identified the seventh symbol." |
The "Jackson" above is played by the young James Spader,
who in an older version currently stars in "The Blacklist."
|
"… the memorable models of science are 'speculative instruments,' — Max Black in Models and Metaphors , Cornell U. Press, 1962 |
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Saturday, April 9, 2016
Expanding the Spielfeld (continued)
"Mr. Conrad was relentless and rigorous in expanding
the parameters of the fields in which he worked."
See also Spielfeld in this journal, as well as Conrad Moonshine.
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Friday, January 15, 2016
Local and Global
The American Mathematical Society today —
|
George E. Weaver (1942-2015) Weaver was a philosophy professor at Bryn Mawr College before retiring in 2008. He had an interest in mathematics, among other fields, and taught discrete mathematics and mathematical logic at Bryn Mawr. A colleague said that Weaver "taught with passion and rigor, and cared deeply about his courses and the students. Students who studied with him had a deep respect and admiration for him." Weaver was an AMS member since 1972. Read more about his life in an obituary published in the Philadelphia Inquirer. |
Weaver reportedly died on December 4, 2015.
Related material in this journal on the date of Weaver's death —
|
Friday, December 4, 2015
Symbology for the Vatican Gift Shop
|
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Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978
“Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
“There is such a thing as harmonic analysis of switching functions.”
— Saying adapted from a young-adult novel
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Sunday, January 10, 2016
Field of Force
The title is adapted from that of George Steiner's book
Fields of Force: Fischer and Spassky at Reykjavik
(Published by Viking Adult on June 25, 1974.)
For fields of narrative force, see the previous post.
See as well a memorable review by the late Florence King
of the novel The Eight by Katherine Neville. An illustration
from that review (The New York Times , January 15, 1989) —
Related material: Closing the Circle (Log24, Sept. 24, 2009).
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Friday, December 25, 2015
At Play in the Fields
See Fields of Force and recent posts.
From PR Newswire in July 2011 —
Campus Crusade for Christ Adopts New Name: Cru
60-year-old Int’l Ministry Aims to Increase
Relevance and Global Effectiveness
Related material:
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Friday, August 14, 2015
Space Station 2015
(A sequel to Space Station 1976)
For Kathleen Gibbons* —
* Note Gibbons's work on "Discrete phase space based on finite fields."
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Wednesday, July 22, 2015
Ein Kampf
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
— Wittgenstein, Philosophical Investigations (1953), Section 109
Related material —
"… the common bond between chess, music, and mathematics
may, finally, be the absence of language."
— George Steiner, Fields of Force: Fischer and Spassky at Reykjavik ,
Viking hardcover, June 1974.
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Sunday, July 12, 2015
Concept Script
"Historically, the idea of a concept script
derives from the Leibnizian project of developing
a so-called 'universal characteristic'
(characteristica universalis )…."
— Dorothea Lotter, "Gottlob Frege: Language,"
in Internet Encyclopedia of Philosophy
Frege as quoted by Lotter —
"Arithmetical, geometrical and chemical symbols
can be regarded as realizations of the Leibnizian
conception in particular fields. The concept script
offered here adds a new one to these – indeed,
the one located in the middle, adjoining all the others."
Wittgenstein —
"Philosophy is a battle against the bewitchment
of our intelligence by means of our language."
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
— Philosophical Investigations (1953), Section 109
Frege, Preface to the Begriffsschrift —
"If it is one of the tasks of philosophy
to break the domination of words over the human spirit
by laying bare the misconceptions
that through the use of language
often almost unavoidably arise
concerning the relations between concepts
and by freeing thought from that with which only
the means of expression of ordinary language,
constituted as they are, saddle it,
then my ideography, further developed for these purposes,
can become a useful tool for the philosopher."
"Wenn es eine Aufgabe der Philosophie ist,
die Herrschaft des Wortes über den menschlichen Geist
zu brechen, indem sie die Täuschungen aufdeckt,
die durch den Sprachgebrauch über die Beziehungen der Begriffe
oft fast unvermeidlich entstehen,
indem sie den Gedanken von demjenigen befreit, womit ihn allein
die Beschaffenheit des sprachlichen Ausdrucksmittels behaftet,
so wird meine Begriffsschrift, für diese Zwecke weiter ausgebildet,
den Philosophen ein brauchbares Werkzeug werden können."
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Wednesday, June 24, 2015
Scoring Plan 9
In keeping with the resurrection themes of the
previous post and of "Plan 9 from Outer Space,"
here is a link to the soundtrack of "Field of Dreams."
Related material:
A post of March 11, 2014, on
truth, cornfields, and Rebecca Goldstein —
Dark Fields of the Republic.
R.I.P., James Horner.
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Wednesday, April 1, 2015
Math’s Big Lies
Two mathematicians, Barry Mazur and Edward Frenkel,
have, for rhetorical effect, badly misrepresented the
history of some basic fields of mathematics. Mazur and
Frenkel like to emphasize the importance of new
research by claiming that it connects fields that previously
had no known connection— when, in fact, the fields were
known to be connected since at least the nineteenth century.
For Mazur, see The Proof and the Lie; for Frenkel, see posts
tagged Frenkel-Metaphors.
See also a story and video on Robert Langlands from the
Toronto Star on March 27, 2015:
"His conjectures are called functoriality and
reciprocity. They made it possible to link up
three branches of math: harmonic analysis,
number theory, and geometry.
To mathematicians, this is mind-blowing stuff
because these branches have nothing to do
with each other."
For a much earlier link between these three fields, see the essay
"Why Pi Matters" published in The New Yorker last month.
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Tuesday, March 24, 2015
Hirzebruch
(Continued from July 16, 2014.)
Some background from Wikipedia:
"Friedrich Ernst Peter Hirzebruch ForMemRS[2]
(17 October 1927 – 27 May 2012)
was a German mathematician, working in the fields of topology,
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'
[3][4][5][6][7][8][9][10][11]"
A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch , a book
reportedly published on September 19, 2012 —
|
"Sadly, on May 27 this year, Friedrich Hirzebruch, Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, |
Also sadly …
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Wednesday, January 21, 2015
High Concept:
The Dark Fields Meet The Big Seal .
Recall the punchline of Tuesday afternoon's post
on the 2012 film "Travelling Salesman" —
"What am I, the farmer's daughter?"
For background from the dark fields of the republic,
see a speech last night by Iowa Senator Joni Ernst.
Related material:
At the end of the 2012 film "Travelling Salesman,"
the main character holds up to the light a letter that has
at the top the presidential seal of the United States:
The camera pans down, and the character then
sees a watermark that echoes another famous seal,
from the U.S. one-dollar bill:
For related paranoia, see the novels of Dan Brown
as well as…
See also Shema and Clocks Striking 13.
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Tuesday, January 20, 2015
Purely Coincidental
"The character and events depicted in this
motion picture are fictitious. Any similarity
to actual persons, living or dead, is purely
coincidental."
— Ending credits of the 2012 film
"Travelling Salesman"
From that film's introduction to the
main character:
"He is presently the Rouse Ball Professor
of Mathematics in the Department of
Pure Mathematics and Mathematical Statistics
at Cambridge University and a fellow at
Trinity College. In 2008… he was awarded
the greatest honor in our profession
when he was presented with the Fields Medal
….
Ladies and gentlemen, it is with great honor
that I now present to you Dr. Timothy Horton."
See also…
A line for a fellow of Trinity:
"What am I, the farmer's daughter?"
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Saturday, December 20, 2014
Dark Fields*
A date in the previous post suggests a flashback to March 11, 2014,
and a post on that date titled "Dark Fields of the Republic"—
This uncredited translation of Plato is, Google Books tells us,
by “Francis MacDonald Cornfield.” The name is an error,
but the error is illuminating —
* See posts mentioning the novel with that title, republished as Limitless.
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Thursday, November 6, 2014
Working Backward
“This setting of the Ave Verum Corpus text was composed
to celebrate the feast of Corpus Christi .” — Wikipedia
“Ave Verum Corpus .”— Madison in the BBC America TV series
“Intruders,” Season 1, Episode 3: “Time Has Come Today.”
See also the Eucharistic meditation of Feb. 13, 2006, linked to in yesterday’s post
on Guy Fawkes Day. (That British holiday originally commemorated the Catholic
Gunpowder Plot of 1605.)
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Wednesday, November 5, 2014
Dark Fields…
From the first of previous Log24 posts tagged “Dark Fields”—
“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:
On the Graphs of Hoffman-Singleton and Higman-Sims.
His truth is marching on.”
See also Foundation Square (October 25, 2014).
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Saturday, October 25, 2014
Foundation Square
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
-
"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and -
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
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Thursday, October 2, 2014
The Christmas Theorem
From the preface to Introduction to the Construction of Class Fields ,
by Harvey Cohn (Cambridge University Press, 1985):
“It is an elementary observation that an integral right triangle
has an even area. Suppose the hypotenuse is prime.
Q. How do we determine from the prime value of the hypotenuse
when the area is divisible by 4, 8, 16, or any higher power of 2?
A. We use class fields constructed by means of transcendental
functions, of course!
The question might have been asked by Pythagoras in about
500 BC….”
The question seems to assume something apparently not known to Pythagoras:
The area is determined uniquely by the prime hypotenuse.
Nontrivial exercise: Prove or disprove this assertion.
Background to the exercise: See Fermat’s Christmas Theorem on the Web,
and a specific remark about prime hypotenuses in a letter from Fermat to
Mersenne on Christmas Day, 1640, quoted in The Mathematical Career
of Pierre de Fermat, 1601-1665 , by Michael Sean Mahoney (Princeton
University Press, 2nd ed,, 1994), pp. 316-317.
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Tuesday, September 30, 2014
Pythagorean Selfie
“Rarely is a TV show as brilliant and as terrible as Selfie .”
— Kevin Fallon on a new ABC TV show that starts tonight at 8 PM ET
A recent selfie from Josefine Lyche’s Instagram page:
For some remarks related to Lyche’s pentagram, see
Lyche + Mathmagic* and also yesterday’s Michaelmas Mystery.
In today’s previous post, the late Harvey Cohn posed a question that
he said might have been asked by Pythagoras:
“It is an elementary observation that an integral right triangle
has an even area. Suppose the hypotenuse is prime.
Q. How do we determine from the prime value of the hypotenuse
when the area is divisible by 4, 8, 16, or any higher power of 2?
A. We use class fields constructed by means of transcendental
functions, of course!”
— From the preface to Introduction to the Construction of Class Fields ,
by Harvey Cohn (Cambridge University Press, 1985)
Illustration:
For a related song, see Prime Suspect (Dec. 13, 2007).
Footnote of 12:14 AM Oct. 1, 2014 —
* That search yields a link to…
This Lyche webpage’s pentagram indicates an interest in Disney rather than
in Satanism. Other Lyche webpages have been less reassuring.
Related material — Posts tagged Elegantly Packaged.
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Pythagoras to Cohn
Harvey Cohn on class field theory and a question that might have
been asked by Pythagoras:
See also yesterday’s Michaelmas Mystery and Michaelmas Texts.
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Thursday, September 11, 2014
A Class by Itself
The American Mathematical Society yesterday:
Harvey Cohn (1923-2014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."
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Tuesday, July 8, 2014
Toward Freedom
A search for “Dark Fields of the Republic,”
an F. Scott Fitzgerald phrase mentioned in
the previous post, yields a book by that title.
“When does a life bend toward freedom?
grasp its direction?”
— Adrienne Rich on page 275 of
Later Poems Selected and New: 1971-2012
The book’s author, Adrienne Rich, died at 82 on
March 27, 2012. See that date in this journal.
See also the following:
| The Diamond Cutters by Adrienne Rich (1955) However legendary, Now, you intelligence Be serious, because Be hard of heart, because Be proud, when you have set |
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Monday, July 7, 2014
“‘Consider,’ said I…”
Roger Cooke in The History of Mathematics: A Brief Course
(2nd ed., Wiley-Interscience, 2005)—
“Like all numbers, the number four is bound to occur
in many contexts.”
— Ch. 1: “The Origin and Prehistory of Mathematics,”
Part 3, “Symbols,” footnote 1, page 11.
As is the number 382:
Click the above image for some related material.
Commentary:
“Once the students are taken in by the story, it will be
the instructor’s job to elaborate on the historical
calculations and proofs.”
— Gary S. Stoudt, Professor of Mathematics,
Indiana U. of Pennsylvania, review of Cooke’s book
at the Mathematical Association of America
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Wednesday, June 18, 2014
A Dark and Stormy Night
This journal on the morning of January 27, 2010,
the day of J. D. Salinger’s death, had a post on
Nietzsche and Heraclitus titled “To Apollo.”
Related material:
“… the wind was noisy the way it is in spooky movies
on the night the old slob with the will gets murdered.”
— From the opening sentence of the first Holden Caulfield
story, published in the Collier’s of December 22, 1945
See also Peter Matthiessen on Zen, Salinger and Vedanta,
and Heraclitus in this journal. Some background—
A quotation from Nietzsche…
(Sämtliche Werke, Kritische Studienausgabe in 15 Bänden (KSA).
Herausgegeben von Giorgio Colli und Mazzino Montinari.
Berlin: De Gruyter, 1980):
“Nietzsche wrote:
‘Seeing the world as a divine game and beyond good and evil:
in this both the Vedanta and Heraclitus are my predecessors.'”
— KSA vol. 11, page 26, as quoted by André van der Braak
in a chapter from his 2011 book Nietzsche and Zen
(Darin, dass die Welt ein göttliches Spiel sei
und jenseits von Gut und Böse —
habe ich die Vedanta-Philosophie
und Heraklit zum Vorgänger.)
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Friday, May 30, 2014
Combinatorial Matrix Classes
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.
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Thursday, March 13, 2014
Entartete Kunst
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) —
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Tuesday, March 11, 2014
Dark Fields of the Republic
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 —
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Sunday, March 9, 2014
At Play in the Fields of Brazil
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 —
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Saturday, March 8, 2014
Women’s History Month
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."
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