Monday, January 17, 2022

Prime Matter

Filed under: General — Tags: Modal Diamond Box — m759 @ 3:17 pm

I an J. Thompson, 7 Dec. 2009—

Quantum mechanics describes the probabilities of actual outcomes in terms of a wave function, or at least of a quantum state of amplitudes that varies with time. The public always asks what the wave function is , or what the amplitudes are amplitudes of . Usually, we reply that the amplitudes are ‘probability amplitudes’, or that the wave function is a ‘probability wave function’, but neither answer is ontologically satisfying since probabilities are numbers , not stuff . We have already rehearsed the objections to the natural world being made out of numbers, as these are pure forms. In fact, ‘waves’, ‘amplitudes’ and ‘probabilities’ are all forms, and none of them can be substances. So, what are quantum objects made of: what stuff ?

According to Heisenberg [6], the quantum probability waves are “a quantitative formulation of the concept of ‘dynamis’, possibility, or in the later Latin version, ‘potentia’, in Aristotle’s philosophy. The concept of events not determined in a peremptory manner, but that the possibility or ‘tendency’ for an event to take place has a kind of reality—a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of the idea or the image—this concept plays a decisive role in Aristotle’s philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature.” Unfortunately Heisenberg does not develop this interpretation much beyond the sort of generality of the above statements, and the concept of ‘potentiality’ remains awkwardly isolated from much of his other thought on this subject [7]. It is unclear even what he means by ‘potentia’.

Reference

Heisenberg, W. 1961 On Modern Physics , London: Orion Press.

Notes

[6] W. Heisenberg, ‘Planck’s discovery and the philosophical problems of atomic physics’, pp. 3-20 in Heisenberg (1961).

[7] Heisenberg, for example, brings into his thought on quantum physics the Kantian phenomena/noumena distinction, as well as some of Bohr’s ideas on ‘complementarity’ in experimental arrangements.

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Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: Gordon Baker, Modal Diamond Box — m759 @ 9:29 am

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

5.5563

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.)

See also—

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 box (square) for necessarily
and (diamond) for possibly. Then, for example, diamond 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.

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Tuesday, August 18, 2009

Tuesday August 18, 2009

Filed under: General,Geometry — Tags: Modal Diamond Box, Sein Feld, Shadow Work — m759 @ 12:00 pm

Prima Materia

(Background: Art Humor: Sein Feld (March 11, 2009) and Ides of March Sermon, 2009)

From Cardinal Manning's review of Kirkman's Philosophy Without Assumptions—

"And here I must confess… that between something and nothing I can find no intermediate except potentia, which does not mean force but possibility."

— Contemporary Review, Vol. 28 (June-November, 1876), page 1017

Furthermore….

Cardinal Manning, Contemporary Review, Vol. 28, pages 1026-1027:

The following will be, I believe, a correct statement of the Scholastic teaching:–

1. By strict process of reason we demonstrate a First Existence, a First Cause, a First Mover; and that this Existence, Cause, and Mover is Intelligence and Power.

2. This Power is eternal, and from all eternity has been in its fullest amplitude; nothing in it is latent, dormant, or in germ: but its whole existence is in actu, that is, in actual perfection, and in complete expansion or actuality. In other words God is Actus Purus, in whose being nothing is potential, in potentia, but in Him all things potentially exist.

3. In the power of God, therefore, exists the original matter (prima materia) of all things; but that prima materia is pura potentia, a nihilo distincta, a mere potentiality or possibility; nevertheless, it is not a nothing, but a possible existence. When it is said that the prima materia of all things exists in the power of God, it does not mean that it is of the existence of God, which would involve Pantheism, but that its actual existence is possible.

4. Of things possible by the power of God, some come into actual existence, and their existence is determined by the impression of a form upon this materia prima. The form is the first act which determines the existence and the species of each, and this act is wrought by the will and power of God. By this union of form with the materia prima, the materia secunda or the materia signata is constituted.

5. This form is called forma substantialis because it determines the being of each existence, and is the root of all its properties and the cause of all its operations.

6. And yet the materia prima has no actual existence before the form is impressed. They come into existence simultaneously;

[p. 1027 begins]

as the voice and articulation, to use St. Augustine's illustration, are simultaneous in speech.

7. In all existing things there are, therefore, two principles; the one active, which is the form– the other passive, which is the matter; but when united, they have a unity which determines the existence of the species. The form is that by which each is what it is.

8. It is the form that gives to each its unity of cohesion, its law, and its specific nature.*

When, therefore, we are asked whether matter exists or no, we answer, It is as certain that matter exists as that form exists; but all the phenomena which fall under sense prove the existence of the unity, cohesion, species, that is, of the form of each, and this is a proof that what was once in mere possibility is now in actual existence. It is, and that is both form and matter.

When we are further asked what is matter, we answer readily, It is not God, nor the substance of God; nor the presence of God arrayed in phenomena; nor the uncreated will of God veiled in a world of illusions, deluding us with shadows into the belief of substance: much less is it catter [pejorative term in the book under review], and still less is it nothing. It is a reality, the physical kind or nature of which is as unknown in its quiddity or quality as its existence is certainly known to the reason of man.

* "… its specific nature"
(Click to enlarge) —

The Catholic physics expounded by Cardinal Manning above is the physics of Aristotle.

For a more modern treatment of these topics, see Werner Heisenberg's Physics and Philosophy. For instance:

"The probability wave of Bohr, Kramers, Slater, however, meant… a tendency for something. It was a quantitative version of the old concept of 'potentia' in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality."

Compare to Cardinal Manning's statement above:

"… between something and nothing I can find no intermediate except potentia…"

To the mathematician, the cardinal's statement suggests the set of real numbers between 1 and 0, inclusive, by which probabilities are measured. Mappings of purely physical events to this set of numbers are perhaps better described by applied mathematicians and physicists than by philosophers, theologians, or storytellers. (Cf. Voltaire's mockery of possible-worlds philosophy and, more recently, The Onion's mockery of the fictional storyteller Fournier's quantum flux. See also Mathematics and Narrative.)

Regarding events that are not purely physical– those that have meaning for mankind, and perhaps for God– events affecting conception, birth, life, and death– the remarks of applied mathematicians and physicists are often ignorant and obnoxious, and very often do more harm than good. For such meaningful events, the philosophers, theologians, and storytellers are better guides. See, for instance, the works of Jung and those of his school. Meaningful events sometimes (perhaps, to God, always) exhibit striking correspondences. For the study of such correspondences, the compact topological space [0, 1] discussed above is perhaps less helpful than the finite Galois field GF(64)– in its guise as the I Ching. Those who insist on dragging God into the picture may consult St. Augustine's Day, 2006, and Hitler's Still Point.

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Saturday, March 7, 2009

Saturday March 7, 2009

Filed under: General,Geometry — Tags: Hexagram 20, Illuminati Stone, Modal Diamond Box — m759 @ 12:00 pm

One or Two Ideas

Today's birthday: Piet Mondrian

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."

The Pope's remarks on that date
were in St. Peter's Square.

From this journal on that date,
a different square —

The Seventh Symbol:

Pictorial version
of Hexagram 20,
Contemplation (View)

The square may be regarded as
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 —

Cover of Mondrian: The Diamond Compositions

— a shape symbolizing
possibility within modal logic
as well as the potentiality of
Aristotle's prima materia.

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Monday, December 29, 2008

Monday December 29, 2008

Filed under: General — Tags: Modal Diamond Box — m759 @ 12:21 pm

The Gift

Robert Stone,
A Flag for Sunrise:

"'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.

Related narratives:

Today's online Times on
the Saturday, Dec. 27,
death of an artist:

Robert Graham obituary, NY Times, 12/29/08

From the Times on the day
of Mr. Graham's death:

"Dale Wasserman… the playwright responsible for two Broadway hits of the 1960s, 'One Flew Over the Cuckoo’s Nest' and 'Man of La Mancha,' died on Sunday [December 21, 2008] at his home in Paradise Valley, Ariz., near Phoenix….

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…"

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Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General — Tags: Modal Diamond Box — m759 @ 12:00 pm

For Balanchine's Birthday

(continued from
January 9, 2003)

George Balanchine

Encyclopædia Britannica Article

born January 22
[January 9, Old Style], 1904,
St. Petersburg, Russia
died April 30, 1983, New York,
New York, U.S.

George Balanchine.
©1983 Martha Swope

original name
Georgy Melitonovich Balanchivadze
most influential choreographer of classical ballet in the United States in the 20th century. His works, characterized by a cool neoclassicism, include The Nutcracker (1954) and Don Quixote (1965), both pieces choreographed for the New York City Ballet, of which he was a founder (1948), the artistic director, and the…

Balanchine, George… (75 of 1212 words)

"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:

"Epiphany signals the traversal
of the finite by the infinite,
of the particular by the universal,
of the mundane by the mystical,
of time by eternity."

— Richard Kearney, 2005,
in The New Arcadia Review

Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.

"… one of the things that worried me about traditional metaphysics, at least as I imbibed it in a very Scholastic manner at University College Dublin in the seventies, is that philosophy was realism and realism was truth. What disturbed me about that was that everything was already acquired; truth was always a systematic given and it was there to be learned from Creation onwards; it was spoken by Jesus Christ and then published by St. Thomas Aquinas: the system as perfect synthesis. Hence, my philosophy grew out of a hunger for the 'possible' and it was definitely a reaction to my own philosophical formation. Yet that wasn't my only reaction. I was also reacting to what I considered to be the deep pessimism, and even at times 'nihilism' of the postmodern turn."

— 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 (

)."

— Keith Allen Korcz

"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)

Comments (4)

Friday, December 29, 2006

Friday December 29, 2006

Filed under: General — Tags: Bullshit Studies, Modal Diamond Box — m759 @ 11:01 am

Tools
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:

"Lewis Wolpert, an eminent developmental biologist at University College London, has just published Six Impossible Things Before Breakfast, a pleasant, though rambling, look at the biological basis of belief. While the book focuses on our ability to form causal beliefs about everyday matters (the wind moved the trees, for example), it spends considerable time on the origins of religious and moral beliefs. Wolpert defends the unusual idea that causal thinking is an adaptation required for tool-making. Religious beliefs can thus be seen as an odd extension of causal thinking about technology to more mysterious matters. Only a species that can reason causally could assert that 'this storm was sent by God because we sinned.' While Wolpert's attitude toward religion is tolerant, he's an atheist who seems to find religion more puzzling than absorbing."

— 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):

"Conductor John Williams, a 2004 Honoree, says, 'Steven, sharing our 34-year collaboration has been a great privilege for me. It's been an inspiration to watch you dream your dreams, nurture them and make them grow. And, in the process, entertain and edify billions of people around the world. Tonight we'd like to salute you, musically, with a piece that expresses that spirit beautifully … It was written by Leonard Bernstein, a 1980 Kennedy Center Honoree who was, incidentally, the first composer to be performed in this hall.' Backed by The United States Army Chorus and The Choral Arts Society, soprano Harolyn Blackwell and tenor Gregory Turay sing the closing number for Spielberg's tribute and the gala itself. It's the finale to the opera 'Candide,' 'Make Our Garden Grow,' and Williams conducts."

— CBS press release

See also the following,
from the conclusion to

"Mathematics and Narrative"

(Log24, Aug. 22, 2005):

"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 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 ()."

— Keith Allen Korcz

"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.

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Tuesday, October 10, 2006

Tuesday October 10, 2006

Filed under: General — Tags: Hash Mate, Modal Diamond Box — m759 @ 8:00 pm

Mate in
Two Seconds

From Oct. 14 last year:

The image “http://www.log24.com/log/pix05B/051014-Tick.gif” cannot be displayed, because it contains errors.

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.'

Notes on Modal Logic:

"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

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Waka waka bang splat
tick tick hash

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.

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Monday, January 23, 2006

Monday January 23, 2006

Filed under: General — Tags: Modal Diamond Box — m759 @ 12:00 pm

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 Diamond of Possibility,

the Baseball Almanac,

and this morning's entry, "Natural Hustler."

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Thursday, January 19, 2006

Thursday, October 13, 2005

Thursday October 13, 2005

Filed under: General — Tags: Modal Diamond Box, My Card, Poem for Pinter — m759 @ 12:00 pm

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.'

Notes on Modal Logic:

"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

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Waka waka bang splat
tick tick hash

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.)

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Monday, August 22, 2005

Monday August 22, 2005

Filed under: General — Tags: Bullshit Studies, Eerdmans Cover, Modal Diamond Box — m759 @ 4:07 pm

The Hole

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."

Part II: Possible Worlds

"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."

"Il faut cultiver notre jardin."– Voltaire

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole

Part III: Modal Theology

"'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)

"We symbolize
logical necessity
with the box (

)
and logical possibility
with the diamond (

)."

— Keith Allen Korcz

"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)

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Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General,Geometry — Tags: bigrock, Modal Diamond Box, Osserman — m759 @ 4:01 pm

Make a Différance

From Frida Saal's
LacanDerrida:

"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names. For that reason their respective works become totally non-superposable and at the same time they were built with an awareness, or at least a partial awareness, of each other. What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."

Presentation

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."

From a text purchased on
Make a Difference Day, Oct. 23, 1999:

The image “http://www.log24.com/log/pix05/050322-Fig39.gif” cannot be displayed, because it contains errors.

22. Without using the Pythagorean Theorem prove that the hypotenuse of an isosceles right triangle will have the length The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.

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 = The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.

— 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.

* See the remarks of Frida Saal above and of Barbara Johnson on mathematics (The Shining of May 29, cited in Readings for St. Patrick's Day).

† 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.

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Thursday, March 3, 2005

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: Bullshit Studies, Cubehenge, Modal Diamond Box, on050220 — m759 @ 2:20 pm

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:

Modal Theology

"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)

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Thursday, February 17, 2005

Tuesday, January 25, 2005

Tuesday January 25, 2005

Filed under: General — Tags: Modal Diamond Box — m759 @ 12:00 pm

Diamonds Are Forever

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

Robert Stone,
A Flag for Sunrise:

" '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.'

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 ()."

— Keith Allen Korcz

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)."

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