The first two pages of a 1989 book by George Steiner—
See also yesterday's posts The Field of the Possible
and Abstraction.
Compare and contrast with Socrates in the Meno
quoting Pindar then discussing with a slave boy
the duplication of the square.
Was Socrates a great philosopher or, as the above
figure seems to indicate and as some say of Steiner,
too clever by half ?
|
A mnemonic from a course titled
“Traditionally, there are two modalities, namely,
|
For less rigorous remarks, search Log24 for Modal Diamond Box.
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 —
"Christmas in February!" — SNL cold open, Feb. 9
Detail, online New York Times front page,
Saint Bridget's Day (Feb. 1), 2013 —
See also Nietzsche and modal logic.
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).
"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) —
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 . . .
(Click to enlarge. Note the infinity favicon.)
" Indeed, one might say that it is possible (ahem ), in another world,
for this article to have been entitled, 'The modal logic of various
set-theoretic multiverse conceptions.' "
"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) —
"Give faith a fighting chance." — Country song
"The proof uses modal logic, which distinguishes
between necessary truths and contingent truths."
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.
The title refers to today's previous post,
"In the Widening Gyre."
"In modal logic, specifically when representing
possible worlds, @ is sometimes used as
a logical symbol to denote the actual world
(the world we are 'at')." —Wikipedia
"What might have been and what has been
Point to one end, which is always present."
— Four Quartets
Click the gyre on the book cover above for further details.
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. |
"I know no writing— except perhaps Henry James's introductory essays— which conveys so clearly and with such an absence of fuss the excitement of the creative artist."
— Graham Greene on A Mathematician's Apology , review in The Spectator , 20 December 1940
"The mere quality and play of an ironic consciousness in the designer left wholly alone, amid a chattering unperceiving world, with the thing he has most wanted to do, with the design more or less realised— some effectual glimpse of that might, by itself, for instance, reward one's experiment."
— Henry James, "Prefaces to the New York Edition," in The Figure in the Carpet and Other Stories, Penguin Books, 1986, with introduction and notes by Frank Kermode
"What? You've found a pattern?"
— Greg Egan, "Wang's Carpets"
See also Notes on Mathematics and Narrative, with its discussion of the tiles of the creative artist Patrick Blackburn in the recent (August 2010) Pythagorean novel The Thousand and the discussion of Wang tiles in Modal Logic, a book from November 2002 whose author also happens to be named Patrick Blackburn.
(Credit for the Greene bibliographic information is due to Janelle Robyn Humphreys, whose doctoral thesis, Shadows of Another Dimension, was published in 2009 by the University of Wollongong.)
|
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."
Pictorial version
of Hexagram 20,
Contemplation (View)
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 —
| 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.”
|
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.
A Little Extra Reading
In memory of
Mary Martin McLaughlin,
a scholar of Heloise and Abelard.
McLaughlin died on June 8, 2006.
"Following the parade, a speech is given by Charles Williams, based on his book The Place of the Lion. Williams explains the true meaning of the word 'realism' in both philosophy and theology. His guard of honor, bayonets gleaming, is led by William of Ockham."
A review by John D. Burlinson of Charles Williams's novel The Place of the Lion:
"… a little extra reading regarding Abelard's take on 'universals' might add a little extra spice– since Abelard is the subject of the heroine's … doctoral dissertation. I'd suggest the article 'The Medieval Problem of Universals' in the online Stanford Encyclopedia of Philosophy."
Michael L. Czapkay, a student of philosophical theology at Oxford:
"The development of logic in the schools and universities of western Europe between the eleventh and fifteenth centuries constituted a significant contribution to the history of philosophy. But no less significant was the influence of this development of logic on medieval theology. It provided the necessary conceptual apparatus for the systematization of theology. Abelard, Ockham, and Thomas Aquinas are paradigm cases of the extent to which logic played an active role in the systematic formulation of Christian theology. In fact, at certain points, for instance in modal logic, logical concepts were intimately related to theological problems, such as God's knowledge of future contingent truths."
The Medieval Problem of Universals, by Fordham's Gyula Klima, 2004:
"… for Abelard, a status is an object of the divine mind, whereby God preconceives the state of his creation from eternity."
(based on Weyl's Symmetry):
"… for then we would know
— 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.
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.)
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)
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
— 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, |
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
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.
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.’ ”
“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….”
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.
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