|
A mnemonic from a course titled
“Traditionally, there are two modalities, namely,
|
For less rigorous remarks, search Log24 for Modal Diamond Box.
Saturday, July 30, 2022
Modal Diamond Box
Sunday, July 10, 2011
Wittgenstein’s Diamond
Philosophical Investigations (1953)—
97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest thing there is
(Tractatus Logico-Philosophicus No. 5.5563).
— Translation by G.E.M. Anscombe
All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.
(Our problems are not abstract but perhaps
the most concrete that there are.)
97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach sein. Sie ist vor aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh. No. 5.5563.)
Related language in Łukasiewicz (1937)—
* Updates of 9:29 PM ET July 10, 2011—
A mnemonic from a course titled “Galois Connections and Modal Logics“—
“Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written 
(square) for necessarily
and 
(diamond) for possibly. Then, for example, P can be read as
‘it is possibly the case that P .'”
| See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—
“The diamond ⋄ symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬⋄¬. The dual symbol □ was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich ‘possible.’” Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159. Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen. Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century. |
Saturday, October 23, 2021
From the Powder-Room of the Muses
The above title is from Northrop Frye —
"Is it possible* that understanding the nature of clarity and order
can cast suspicion on the very ideas of clarity and order?"
— Douglas Sadao Aoki, University of Alberta, "The Thing Never
Speaks for Itself: Lacan and the Pedagogical Politics of Clarity,"
Harvard Educational Review , Vol. 70, No. 3, Fall 2000,
Copyright © by President and Fellows of Harvard College.
Related scholarly citation by Aoki —
The cited source:
* For the diamond as a symbol of possibility , see modal diamond box .
Saturday, January 18, 2020
Interplay
"This interplay of necessity and contingency
produces our anxious— and highly pleasurable—
speculation about the future path of the story."
— Michel Chaouli in "How Interactive Can Fiction Be?"
(Critical Inquiry 31, Spring 2005, page 613.)
See also . . .
Continuing previous Modal Diamond Box posts:
Tuesday, April 10, 2018
Logos
Sunday, February 4, 2018
Logos for Sunday, February 4
"The walls in the back of the room show geometric shapes
that remind us of the logos on a space shuttle. "
— Web page on an Oslo art installation by Josefine Lyche.
See also Subway Art posts.
The translation above was obtained via Google.
The Norwegian original —
"På veggene bakerst i rommer vises geometriske former
som kan minne om logoene på en romferge."
Related logos — Modal Diamond Box in this journal:
Logos for Philosophers
(Suggested by Modal Logic) —
Friday, February 21, 2014
Night’s Hymn of the Rock
One way of interpreting the symbol 
at the end of yesterday's post is via
the phrase "necessary possibility."
See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .
The Tablet post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—
On that day (July 23) another weblog had
a post titled
Wallace Stevens: Night's Hymn of the Rock.
Some related narrative —
I prefer the following narrative —
Part I: Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"
Part II: Mystery Box III: Inside, Outside (2014)
Sunday, November 10, 2013
Abstraction
(Continued from Dec. 6, 2012)
Context:
Chinese Field and Modal Diamond .
(See also today’s previous post.)
Saturday, July 20, 2024
Hillbilly Geometry
For Captain Maybelline*
"Your life, little girl, is an empty page . . . ." — Song lyric
* A reference to a New Yorker cartoon of July 18, 2024.
For Captain Maybelline*
Thursday, December 28, 2023
Basque Country Art Book
Book description at Amazon.com, translated by Google —
|
Las matemáticas como herramienta
Mathematics as a tool by Raúl Ibáñez Torres Kindle edition in Spanish, 2023 Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares). |
From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —
Publisher: Los Libros de La Catarata (October 24, 2023)
Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez
(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)
About Ibáñez (translated from Amazon.com by Google):
Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).
Thursday, December 14, 2023
Unsocratic Dialogue (with Gemini AI in Bard Today)
|
What is the Cullinane diamond theorem?
|
Saturday, October 14, 2023
Review
|
Robert Stone " 'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.' 'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.' Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion. 'Hey,' he said to the priest, 'diamonds are forever! You heard of that, right? That means something, don't it?'
'I have heard it,' Egan said. 'Perhaps it has a religious meaning.' "
"We symbolize logical necessity — Keith Allen Korcz |
Saturday, March 12, 2022
Logos and Branding
The "branding" part of this post's title and tag —
|
The scene went from bad to worse. The camerlengo’s torn cassock, having been only laid over his chest by Chartrand, began to slip lower. For a moment, Langdon thought the garment might hold, but that moment passed. The cassock let go, sliding off his shoulders down around his waist. The gasp that went up from the crowd seemed to travel around the globe and back in an instant. Cameras rolled, flashbulbs exploded. On media screens everywhere, the image of the camerlengo’s branded chest was projected, towering and in grisly detail. Some screens were even freezing the image and rotating it 180 degrees. The ultimate Illuminati victory. Langdon stared at the brand on the screens. Although it was the imprint of the square brand he had held earlier, the symbol now made sense. Perfect sense. The marking’s awesome power hit Langdon like a train. Orientation. Langdon had forgotten the first rule of symbology. When is a square not a square? He had also forgotten that iron brands, just like rubber stamps, never looked like their imprints. They were in reverse. Langdon had been looking at the brand’s negative ! As the chaos grew, an old Illuminati quote echoed with new meaning: ‘A flawless diamond, born of the ancient elements with such perfection that all those who saw it could only stare in wonder.’ Langdon knew now the myth was true. Earth, Air, Fire, Water. The Illuminati Diamond. — Dan Brown, Angels & Demons |
I prefer Modal Nietzsche.
Saturday, April 10, 2021
Possibility
The previous post, on a Joyce symposium in
Utrecht on June 15-20, 2014, suggests a review
of this journal in June 2014. From June 21
of that year —
"Without the possibility that
an origin can be lost, forgotten,
or alienated into what springs
forth from it, an origin could
not be an origin. The possibility
of inscription is thus a necessary
possibility, one that must always
be possible."
— Page 157 of The Tain of the Mirror:
Derrida and the Philosophy of Reflection ,
by Rodolphe Gasché, Harvard U. Press, 1986
Related art suggested by the above modal logic —
Saturday, July 21, 2018
Building-Block Theory
(A sequel to yesterday’s Geometry for Jews)
From this journal on the above UCI posting date — April 6, 2018 —
From this journal on the above lecture date — April 26, 2018 —
illustrations in a post titled Defining Form —
For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.
Friday, April 6, 2018
A Service
From a Boston Globe obituary for Andrew Lewis, an Oscar-nominated
screenwriter who reportedly died at 92 on Feb. 28, 2018 —
"A service has been held for Mr. Lewis . . . ."
— Bryan Marquard, Globe staff, April 5, 2018
From this journal on the reported date of his death —
The Globe reports that Lewis's father was Clarence Irving Lewis,
a professor of philosophy at Harvard University.
Fact check: See page 246 of C. I. Lewis: The Last Great Pragmatist ,
by Murray G. Murphey (SUNY Press, 2005).
Figure (a) above is not unrelated to philosophy. See Plato 's Meno dialogue.
See also a different diamond — a symbol devised by C. I. Lewis for use in
modal logic — in the post Wittgenstein's Diamond (July 10, 2011).
Tuesday, December 12, 2017
Google Sutra
Suggested by a Diamond Sutra webpage, by a recent Log24 post . . .
Logos for Philosophers
(Suggested by Modal Logic) —
. . . and by the Google Play Store logo —
For further details, see . . .
Friday, December 8, 2017
Logos (Continued)
"Denn die Welt braucht ewig die Wahrheit,
also braucht sie ewig Heraklit:
obschon er ihrer nicht bedarf.
Was geht ihn sein Ruhm an?
Der Ruhm bei »immer fortfließenden Sterblichen!«,
wie er höhnisch ausruft.
Sein Ruhm geht die Menschen etwas an, nicht ihn,
die Unsterblichkeit der Menschheit braucht ihn,
nicht er die Unsterblichkeit des Menschen Heraklit.
Das, was er schaute, die Lehre vom Gesetz im Werden
und vom Spiel in der Notwendigkeit , muß von jetzt
ab ewig geschaut werden: er hat von diesem größten
Schauspiel den Vorhang aufgezogen."
Logos for Philosophers
(Suggested by Modal Logic) —
Thursday, February 20, 2014
Relativity Blues
A review of this date in 2005 —
Modal Theology
“We symbolize logical necessity
with the box (
)
and logical possibility
with the diamond (
).”
— Keith Allen Korcz
And what do we
symbolize by ?
Friday, February 1, 2013
Get Quotes
For Tony Kushner fans:
For logic fans:
In the box-diamond notation, the axiom Searle quotes is
.
"The euclidean property guarantees the truth of this." — Wikipedia
Linking to Euclid
Clicking on "euclidean" above yields another Wikipedia article…
"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."
Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.
Saturday, March 7, 2009
Saturday March 7, 2009
One or Two Ideas
|
From James Joyce's A Portrait of the Artist as a Young Man:
he hearth and began to stroke his chin. –When may we expect to have something from you on the esthetic question? he asked. –From me! said Stephen in astonishment. I stumble on an idea once a fortnight if I am lucky. –These questions are very profound, Mr Dedalus, said the dean. It is like looking down from the cliffs of Moher into the depths. Many go down into the depths and never come up. Only the trained diver can go down into those depths and explore them and come to the surface again. –If you mean speculation, sir, said Stephen, I also am sure that there is no such thing as free thinking inasmuch as all thinking must be bound by its own laws. –Ha! –For my purpose I can work on at present by the light of one or two ideas of Aristotle and Aquinas. –I see. I quite see your point. |
Besides being Mondrian's birthday, today is also the dies natalis (in the birth-into-heaven sense) of St. Thomas Aquinas and, for those who believe worthy pre-Christians also enter heaven, possibly of Aristotle.
Pope Benedict XVI explained the dies natalis concept on Dec. 26, 2006:
"For believers the day of death, and even more the day of martyrdom, is not the end of all; rather, it is the 'transit' towards immortal life. It is the day of definitive birth, in Latin, dies natalis."
were in St. Peter's Square.
Pictorial version
of Hexagram 20,
Contemplation (View)
symbolizing art itself.
(See Nov.30 – Dec.1, 2008.)
In honor of
Aristotle and Aquinas,
here is a new web site,
illuminati-diamond.com,
with versions of the diamond shape
made famous by Mondrian —
— a shape symbolizing
possibility within modal logic
as well as the potentiality of
Aristotle's prima materia.
Monday, December 29, 2008
Monday December 29, 2008
Robert Stone,
"'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.' Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion." For further details, click on the diamond. |
Today's online Times on
the Saturday, Dec. 27,
death of an artist:
Mr. Wasserman wrote more than 75 scripts for television, the stage and the movies, including screenplays for 'The Vikings' (1958), a seafaring epic with Tony Curtis and Kirk Douglas, and 'A Walk With Love and Death' (1969), a John Huston film set in 14th-century Europe….
He feuded with… John Huston, who gave the lead female role in 'Walk' to his teenage daughter, Anjelica, against Mr. Wasserman's wishes. And he never attended ceremonies to receive the awards he won."
Accepting for Mr. Wasserman:
Mr. Graham's widow,
Anjelica Huston —
"Well…"
Tuesday, January 23, 2007
Tuesday January 23, 2007
“The most prominent critic
of the modal notions is Quine.
Throughout his career, he has
argued against the use of notions
like necessity and possibility.”
— Michael J. Loux,
Note 1 of Chapter 5,
“The Necessary and the Possible,”
in Metaphysics:
A Contemporary Introduction
(Routledge, second edition,
January 1, 2002)
“Personality is a synthesis of
possibility and necessity.”
— Soren Kierkegaard,
The Sickness Unto Death
Related material:
Plato, Pegasus,
and the Evening Star
Diamonds Are Forever
Dream a Little Dream
Update of 3:45 PM:
From Arts & Letters Daily
this afternoon–
“Existentialism is not all gloom,
even if Heidegger looks pretty sour
in those photos. It’s a philosophy
that America needs now, says
the late Robert Solomon…
more … obit”
See also Jan. 2,
the date of
Solomon’s death
in Switzerland,
and click on the
following symbol
from that date:
Tuesday, January 9, 2007
Tuesday January 9, 2007
(private, cut from prev. entry)
| The diamond is used in modal logic to symbolize possibility. |
The 3×3 grid may also be used
to illustrate “possibility.” It leads,
as noted at finitegeometry.org, to
the famed “24-cell,” which may be
pictured either as the diamond
figure from Plato’s Meno —
Click for details.
— or as a figure
with 24 vertices:
Click for details.
The “diamond” version of the
24-cell seems unrelated to the
second version that shows all
vertices and edges, yet the
second version is implicit,
or hidden, in the first.
Hence “possibility.”
Neither version of the 24-cell
seems related in any obvious
way to the 3×3 grid, yet both
versions are implicit,
or hidden, in the grid.
Hence “possibility.”
Tuesday January 9, 2007
(continued from
January 9, 2003)
George Balanchine
|
"What on earth is
a concrete universal?"
— Robert M. Pirsig
Review:
From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:
"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."
There is, however,
"the field of reason"–
the 3×3 grid:
Click on grid
for details.
As Rosalind Krauss
has noted, some artists
regard the grid as
"a staircase to
the Universal."
Other artists regard
Epiphany itself as an
approach to
the Universal:
— Richard Kearney, 2005,
in The New Arcadia Review
Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.
— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 2005-2006
For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:
|
"We symbolize
logical necessity with the box )and logical possibility with the diamond )."
"The possibilia that exist,
— Michael Sudduth, |
Friday, December 29, 2006
Friday December 29, 2006
of Christ Church
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon
Click on picture for details.
Today is the feast
of St. Thomas Becket.
In his honor, a meditation
on tools and causation:
— Review by H. Allen Orr in
The New York Review of Books,
Vol. 54, No. 1, January 11, 2007
"An odd extension"–
Wolpert's title is, of course,
from Lewis Carroll.
Related material:
"It's a poor sort of memory
that only works backwards."
— Through the Looking-Glass
An event at the Kennedy Center
broadcast on
December 26, 2006
(St. Steven's Day):
from the conclusion to
(Log24, Aug. 22, 2005):
"At times, bullshit can
only be countered
with superior bullshit."
— Norman Mailer
"The concept of possible worlds dates back to at least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
"Il faut cultiver notre jardin."
— Voltaire
"We symbolize
logical necessity
with the box )
and logical possibility
with the diamond )."
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity,
Christ Church College, Oxford
(the home of Lewis Carroll)
For further details,
click on the
Christ Church diamond.
Tuesday, October 10, 2006
Tuesday October 10, 2006
Two Seconds
From Oct. 13 last year
(Yom Kippur):
|
A Poem for Pinter
Oct. 13, 2005 The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature: "Earlier this year, he announced his decision to retire from playwriting in favour of poetry," Michael Muskal in today's Los Angeles Times: "Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits." Robert Stone, A Flag for Sunrise (See Jan. 25): "'That old Jew gave me this here.' Egan looked at the diamond…. 'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal….'"
"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth. The words 'necessary' and 'possible' are called modal connectives, or modalities. A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true. In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"
Commentary:
"Waka" also means Japanese poem or Maori canoe. (For instance, this Japanese poem and this Maori canoe.)
For a meditation on "bang splat," see Sept. 25-29. For the meaning of "tick tick," see Emily Dickinson on "degreeless noon." "Hash," of course, signifies "checkmate." (See previous three entries.) |
For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.
That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.
For more on the former, see
Death in Two Seconds.
For more on the latter, see
The Halmos Tombstone.
Monday, January 23, 2006
Monday January 23, 2006
The Case
An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:
From The Hustler, by Walter Tevis:
The only light in the room was from the lamp over the couch where she was reading.
He looked at her face. She was very drunk. Her eyes were swollen, pink at the corners. "What's the book?" he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
She blinked up at him, smiled sleepily, and said nothing.
"What's the book?" His voice had an edge now.
"Oh," she said. "It's Kierkegaard. Soren Kierkegaard." She pushed her legs out straight on the couch, stretching her feet. Her skirt fell back a few inches from her knees. He looked away.
"What's that?" he said.
"Well, I don't exactly know, myself." Her voice was soft and thick.
He turned his face away from her again, not knowing what he was angry with. "What does that mean, you don't know, yourself?"
She blinked at him. "It means, Eddie, that I don't exactly know what the book is about. Somebody told me to read it, once, and that's what I'm doing. Reading it."
He looked at her, tried to grin at her– the old, meaningless, automatic grin, the grin that made everybody like him– but he could not. "That's great," he said, and it came out with more irritation than he had intended.
She closed the book, tucked it beside her on the couch. "I guess this isn't your night, Eddie. Why don't we have a drink?"
"No." He did not like that, did not want her being nice to him, forgiving. Nor did he want a drink.
Her smile, her drunk, amused smile, did not change. "Then let's talk about something else," she said. "What about that case you have? What's in it?" Her voice was not prying, only friendly. "Pencils?"
"That's it," he said. "Pencils."
She raised her eyebrows slightly. Her voice seemed thick. "What's in it, Eddie?"
"Figure it out yourself." He tossed the case on the couch.
Related material:
Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,
the Baseball Almanac,
and this morning's entry, "Natural Hustler."
Thursday, January 19, 2006
Thursday January 19, 2006
Saturday, January 14, 2006
Saturday January 14, 2006
(See previous entry.)
— Brian Weatherson, Associate Professor of Philosophy, Cornell University, May 11, 2004
Here, on the other hand, is a way of framing the problem that is entirely idiosyncratic:
On this date:
Probability:
In 1970, William Feller died.
Modality:
In 1978, Kurt Gödel died.
Intersection:
In 1898, the Rev. Deacon Charles Lutwidge Dodgson died.
Thursday, October 13, 2005
Thursday October 13, 2005
A Poem for Pinter
The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:
"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"
Michael Muskal in today's Los Angeles Times:
"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."
Robert Stone, A Flag for Sunrise (See Jan. 25):
"'That old Jew gave me this here.' Egan looked at the diamond…. 'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal….'"
"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth. The words 'necessary' and 'possible' are called modal connectives , or modalities . A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true. In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"
|
A Poem for Pinter
|
Commentary:
"Waka" also means Japanese poem or Maori canoe.
(For instance, this Japanese poem and this Maori canoe.)
For a meditation on "bang splat," see Sept. 25-29.
For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."
"Hash," of course, signifies "checkmate." (See previous three entries.)
Monday, August 22, 2005
Monday August 22, 2005
Part I: Mathematics and Narrative
Apostolos Doxiadis on last month's conference on "mathematics and narrative"–
Doxiadis is describing how talks by two noted mathematicians were related to
"… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "
Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. [In one part] I’ll tell you a few things about proofs. [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’ But the similarity was not clearly explained: the hole, you see, the ‘miracle.’ Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer
Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
Background:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)
'…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
— Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)
"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."
|
"We symbolize
logical necessity with the box )and logical possibility with the diamond ).
"The possibilia that exist,
— Michael Sudduth, |
Tuesday, May 31, 2005
Tuesday May 31, 2005
Subject and Predicates
“A Chu space is a set X of subjects and a set A of predicates on those subjects. These stand in a symbiotic relationship in which the nature of each is determined by the other. Each subject is characterized by the values the predicates take on it, while each predicate is characterized by its values on subjects.”
— Vaughan Pratt, Chu Spaces
Click here for Sambin’s paper (ps).
It would seem that Pratt and Sambin need to reconcile their similar predicates for the same subject.
For some background on Sambin’s approach to the subject, see
A View of its Evolution (pdf),
by Robert Goldblatt at
Victoria University of Wellington’s
Centre for Logic, Language,
and Computation
Information Transfer
Across Chu Spaces (pdf),
by Johan van Benthem
at the University of Amsterdam’s
Institute for Logic, Language,
and Computation
For a gloss on Sambin’s words
see the Log24 entry of Epiphany, 2005.
Tuesday, March 22, 2005
Tuesday March 22, 2005
Make a Différance
From Frida Saal's
Lacan 
Derrida:
From a Contemporary Literary Theory website:
"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."
Make a Difference Day, Oct. 23, 1999:
 22. Without using the Pythagorean Theorem prove that the hypotenuse of an isosceles right triangle will have the length  if the equal legs have the length 1. Suggestion: Consider the similar triangles in Fig. 39.23. The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas. We cannot do so now because we have not yet considered the idea of area. Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = .
— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941) |
Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics* — they may in other respects be quite superior to men:
The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.
† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005. See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.
Thursday, March 3, 2005
Thursday March 3, 2005
Matrix group actions,
March 26, 1985
"We symbolize logical necessity
with the box )
and logical possibility
with the diamond ).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Sunday, February 20, 2005
Sunday February 20, 2005
Relativity Blues
Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry. Here is some related material.
In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).
(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)
On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:
"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."
One possible response to such statements was recently provided in some detail by a Princeton philosophy professor. See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.
A more thoughtful response would take into account the following:
1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson
2. The physics underlying Einstein's remarks on free will, God, and dice
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels Secret Passages and Broken Symmetries
4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University
5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.
From a review of Gilead by Jane Vandenburgh:
"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"
For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries. If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003, and those of November 5, 2002.
From Many Dimensions (Eerdmans paperback, 1963, page 53):
"Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?"
A recent answer:
"We symbolize logical necessity
with the box )
and logical possibility
with the diamond ).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Thursday, February 17, 2005
Thursday February 17, 2005
"We symbolize logical necessity
with the box )
and logical possibility
with the diamond ).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by ?
On the Lapis Philosophorum,
the Philosophers' Stone –
"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"
– Many Dimensions,
by Charles Williams, 1931
(Eerdmans paperback,
April 1979, pp. 43-44)
"The lapis was thought of as a unity
and therefore often stands for
the prima materia in general."
– Aion, by C. G. Jung, 1951
(Princeton paperback,
1979, p. 236)
"Its discoverer was of the opinion that
he had produced the equivalent of
the primordial protomatter
which exploded into the Universe."
– The Stars My Destination,
by Alfred Bester, 1956
(Vintage hardcover,
July 1996, p. 216)
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
See also
The Diamond Archetype.
For more on modal theology, see
Kurt Gödel's Ontological Argument
and
The Ontological Argument
from Anselm to Gödel.
Tuesday, January 25, 2005
Tuesday January 25, 2005
Diamonds Are Forever
" 'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.'
Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion.
'Hey,' he said to the priest, 'diamonds are forever! You heard of that, right? That means something, don't it?'
'I have heard it,' Egan said. 'Perhaps it has a religious meaning.' "
"We symbolize logical necessity
with the box )
and logical possibility
with the diamond ).
From
DIALECTIC AND EXISTENCE
IN KIERKEGAARD AND KANT
Nythamar Fernandes de Oliveira
Pontifical Catholic University
at Porto Alegre, Brazil
"Such is the paradoxical 'encounter' of the eternal with the temporal. Just like the Moment of the Incarnation, when the Eternal entered the temporal, Kierkegaard refers to the category of the Instant (Danish Ojeblikket, 'a glance of the eye, eyeblink,' German Augenblick) as the dialectical kernel of our existential consciousness:
If the instant is posited, so is the eternal –but also the future, which comes again like the past … The concept around which everything turns in Christianity, the concept which makes all things new, is the fullness of time, is the instant as eternity, and yet this eternity is at once the future and the past.
Although I cannot examine here the Kierkegaardian conception of time, the dialectical articulation of time and existence, as can be seen, underlies his entire philosophy of existence, just as the opposition between 'eternity' and 'temporality': the instant, as 'an atom of eternity,' serves to restructure the whole synthesis of selfhood into a spiritual one, in man’s 'ascent' toward its Other and the Unknown. In the last analysis, the Eternal transcends every synthesis between eternity and time, infinity and finiteness, preserving not only the Absolute Paradox in itself but above all the wholly otherness of God. It is only because of the Eternal, therefore, that humans can still hope to attain their ultimate vocation of becoming a Chistian. As Kierkegaard writes in Works of Love (1847),
The possibility of the good is more than possibility, for it is the eternal. This is the basis of the fact that one who hopes can never be deceived, for to hope is to expect the possibility of the good; but the possibility of the good is eternal. …But if there is less love in him, there is also less of the eternal in him; but if there is less of the eternal in him, there is also less possibility, less awareness of possibility (for possibility appears through the temporal movement of the eternal within the eternal in a human being)."
Monday, January 24, 2005
Monday January 24, 2005
The Diamond
of Possibility
"We symbolize logical necessity with the box )).p = ~~pp = ~~p.
And what do we
symbolize by ?
Monday January 24, 2005
Old School Tie
“We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates’ yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, ‘There must be a formula for how ugly your tie is.’ ”
have been extremely influential.”– Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke
“Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections……. diamonds and boxes are upper and lower adjoints of Galois connections….”
and Applications: A Survey”
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
Galois Correspondence
Evariste Galois
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962
For metaphor and
algebra combined, see
“Symmetry invariance
in a diamond ring,”
A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.