… and Arthur Koestler
|
The theme of the January 2010 issue of the
Notices of the American Mathematical Society was “Mathematics and the Arts.”
Related material:
|
See also two posts from the day Peter Jennings died—
Wednesday, June 8, 2011
For Saint Peter
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Monday, December 27, 2010
Church Diamond
Also known, roughly speaking, as confluence or the Church-Rosser property.
From “NYU Lambda Seminar, Week 2” —
[See also the parent page Seminar in Semantics / Philosophy of Language or:
What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn’t Know To Ask)]
A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.
The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn’t matter which order the sub-expressions are evaluated in.
A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn’t terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.
But the untyped lambda calculus enjoys some compensation for this weakness. It’s Turing complete! It can represent any computation we know how to describe. (That’s the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don’t know how to rigorously define “any computation we know how to describe.”) And in fact, it’s been proven that you can’t have both. If a computational system is Turing complete, it cannot be strongly normalizing.
There is no connection, apart from the common reference to an elementary geometric shape, between the use of “diamond” in the above Church-Rosser sense and the use of “diamond” in the mathematics of (Cullinane’s) Diamond Theory.
Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.
Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —
Link One — “Insane Symmetry” (Click image for further details)—
See also the quilt symmetry in this journal on Christmas Day.
Link Two — Divine Symmetry
(George Steiner on the Name in this journal on Dec. 31 last year (“All about Eve“)) —
“The links are direct between the tautology out of the Burning Bush, that ‘I am’ which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That ‘I am’ has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes ‘a dead letter.'”
– George Steiner, Grammars of Creation
(See also, from Hanukkah this year, A Geometric Merkabah and The Dreidel is Cast.)
Link Three – Spanning the Arc —
Part A — Architect Louis Sullivan on “span” (see also Kindergarten at Stonehenge)
Part B — “Span” in category theory at nLab —
Also from nLab — Completing Spans to Diamonds
“It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property).”
The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).
They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).
For more on what the above San Francisco art curator is pleased to call “insane symmetry,” see this journal on Christmas Day.
For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times “The Stone” essay “Span: A Remembrance,” from December 22—
“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”
Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”
See also the link from Christmas Day to remarks on Euclid and “architectonic” in Mere Geometry.
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Monday, September 27, 2010
The Social Network…
… In the Age of Citation
1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).
— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology, Vol. 23, pp. 127–146 (1993)
From this paper's CiteSeer page—
Citations
| 766 | Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994 |
| 100 | The act of creation – Koestler – 1964 |
| 75 | Visual Thinking – Arnheim – 1969 |
Visual Image of the Problem—
From a Google search today:
Related material—
"It is better to light one candle…"
"… the early favorite for best picture at the Oscars" — Roger Moore
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Wednesday, June 16, 2010
Brightness at Noon
David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —
See also this morning's post as well as
Monday's post quoting George David Birkhoff —
"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."
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Sunday, January 17, 2010
Welcome to the Ape Stuff
NY Times obituary of Knox Burger,
book editor and agent, who died at 87 on January 4—
"As a magazine editor in the 1950s, Knox Burger published Kurt Vonnegut’s first short story….
During Mr. Burger’s tenure at Collier’s, a short story by Vonnegut, whom he had known slightly when both were at Cornell and who was then working in public relations for General Electric, crossed his desk. He asked for changes, which Vonnegut made, and the story, 'Report on the Barnhouse Effect,' appeared in the magazine in February 1950. It was the first published work of fiction for Vonnegut, who recounted the episode decades later….
At least half a dozen authors… honored Mr. Burger by dedicating books to him. Vonnegut, who died in 2007, did, too. His dedication of Welcome to the Monkey House, a 1968 collection of short stories that included 'Report on the Barnhouse Effect,' read:
'To Knox Burger. Ten days older than I am. He has been a very good father to me.'"
A Jesuit at the
Gerard Manley Hopkins Archive—
"Bisociation": The Act of Creation
"Koestler’s concept of ‘bisociation’… enters into the very ‘act of creation.’ In every such act, writes Koestler, the creator ‘bisociates,’ that is, combines, two ‘matrices’– two diverse patterns of knowing or perceiving– in a new way. As each matrix carries its own images, concepts, values, and ‘codes,’ the creative person brings together– ‘bisociates’– two diverse matrices not normally connected."
– Joseph J. Feeney, S.J.
Robert Stone in A Flag for Sunrise
(Knopf hardcover, 1981)–
"The eye you see him with is the same eye with which he sees you."
– Father Egan on page 333
Pablo on page 425–
"'…You know, he told me– that old man told me– the eye you look at it with, well, that's the eye it sees you with. That's what he told me.'
Holliwell was moved to recall an experiment he had once read about; he had clipped the report of it for his class. An experimenter endeavoring to observe chimpanzee behavior had fashioned a spy hole in the door of the animals' chamber through which he might watch them unobserved. Putting his eye to it, he had seen nothing more than what he finally identified as the eye of a chimpanzee on the other side of the door. Ape stuff."
More ape stuff from a Jesuit–
"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond,
Is immortal diamond."
— Gerard Manley Hopkins,
"That Nature is a Heraclitean Fire
and of the comfort of the Resurrection"
More ape stuff from myself–
Problem: Perform this transformation
by combining the sorts of permutations allowed
in the diamond puzzle. A solution: click here.
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Tuesday, January 5, 2010
Artifice of Eternity
A Medal
In memory of Byzantine scholar Ihor Sevcenko,
who died at 87 on St. Stephen's Day, 2009–
| William Grimes on Sevcenko in this morning's New York Times:
"Perhaps his most fascinating, if uncharacteristic, literary contribution came shortly after World War II, when he worked with Ukrainians stranded in camps in Germany for displaced persons. In April 1946 he sent a letter to Orwell, asking his permission to translate 'Animal Farm' into Ukrainian for distribution in the camps. The idea instantly appealed to Orwell, who not only refused to accept any royalties but later agreed to write a preface for the edition. It remains his most detailed, searching discussion of the book." |
See also a rather different medal discussed
here in the context of an Orwellian headline from
The New York Times on Christmas morning,
the day before Sevcenko died.
That headline, at the top of the online front page,
was "Arthur Koestler, Man of Darkness."
The medal, offered as an example of brightness
to counteract the darkness of the Times, was designed
by Leibniz in honor of his discovery of binary arithmetic.
See Brightness at Noon and Brightness continued.
"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118
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Wednesday, December 30, 2009
Fearful Symmetry
Arthur Koestler by David Levine,
New York Review of Books,
December 17, 1964
| A Jesuit at the Gerard Manley Hopkins Archive: ‘Bisociation’: The Act of Creation Koestler’s concept of ‘bisociation’… enters into the very ‘act of creation.’ In every such act, writes Koestler, the creator ‘bisociates,’ that is, combines, two ‘matrices’– two diverse patterns of knowing or perceiving– in a new way. As each matrix carries its own images, concepts, values, and ‘codes,’ the creative person brings together– ‘bisociates’– two diverse matrices not normally connected. — Joseph J. Feeney, S.J. |
See also December 9, 2009:
The theme of the January 2010 issue of the
Notices of the American Mathematical Society
is “Mathematics and the Arts.”
Related material:
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Monday, December 28, 2009
Brightness at Noon, continued
This journal’s Christmas Day entry, Brightness at Noon, was in response to the Orwellian headline “Arthur Koestler, Man of Darkness,” at the top of the online New York Times front page on Christmas morning.
The entry offered, as an example of brightness, some thoughts of Leibniz on his discovery of binary arithmetic.
Related material:
“To make all things from nothing, unity suffices.” So it is written on a medal entitled Imago Creationis and designed by Leibniz to “exhibit to posterity in silver” his discovery of the binary system. Baron Gottfried Wilhelm von Leibniz (also Leibnitz) 1646-1716. Philosopher and mathematician. Invented calculus independently of Newton. Proposed the metaphysical theory that we live in “the best of all possible worlds.” He also discovered binary number system and believed in its profound metaphysical significance. He noticed similarity with the ancient Chinese divination system “I Ching.” We chose him for our patron, for Krawtchuk polynomials can be understood as a sophistication of the simple counting of 0 and 1… — Philip Feinsilver and Jerzy Kocik, 17 July 2001 |
From Mikhail Krawtchouk: Short Biography—
Anyone knowing even a little Soviet history of the thirties can conclude that Krawtchouk could not avoid the Great Terror. During the Orwellian “hours of hatred” in 1937 he was denounced as a “Polish spy,” “bourgeois nationalist,” etc. In 1938, he was arrested and sentenced to 20 years of confinement and 5 years of exile.
Academician Krawtchouk, the author of results which became part of the world’s mathematical knowledge, outstanding lecturer, member of the French, German, and other mathematical societies, died on March 9, 1942, in Kolyma branch of the GULAG (North-Eastern Siberia) more than 6 months short of his 50th birthday.
Incidentally, happy birthday
to John von Neumann.
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Friday, December 25, 2009
Brightness at Noon
New York Times online front page
Christmas morning:
"Arthur Koestler, Man of Darkness"–
The photo is of Koestler in 1931 on a zeppelin expedition to the North Pole.
"The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ’see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”
– Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
From Opus Postumum by Immanuel Kant, Eckart Förster, Cambridge U. Press, 1995, p. 260:
"In January 1697, Leibniz accompanied his New Year Congratulations to Rudolf August with the design of a medal with the duke's likeness on one side, and the 'image of Creation' in terms of the binary number system on the other. Concerning the inscription on this side, Leibniz writes: 'I have thought for a while about the Motto dell'impresa and finally have found it good to write this line: omnibus ex nihilo ducendis SUFFICIT UNUM [To make all things from nothing, UNITY SUFFICES], because it clearly indicates what is meant by the symbol, and why it is imago creationis' (G. F. Leibniz, Zwei Briefe über das binäre Zahlensystem und die chinesische Philosophie, ed. Renate Loosen and Franz Vonessen, Chr. Belser Verlag: Stuttgart 1968, p. 21)."
Figure from Rudolf Nolte’s
Gottfried Wilhelms Baron von Leibniz
Mathematischer Beweis der Erschaffung und
Ordnung der Welt in einem Medallion…
(Leipzig: J. C. Langenheim, 1734).
"And so that I won’t come entirely empty-handed this time, I enclose a design of that which I had the pleasure of discussing with you recently. It is in the form of a memorial coin or medallion; and though the design is mediocre and can be improved in accordance with your judgment, the thing is such, that it would be worth showing in silver now and unto future generations, if it were struck at your Highness’s command. Because one of the main points of the Christian Faith, and among those points that have penetrated least into the minds of the worldly-wise and that are difficult to make with the heathen is the creation of all things out of nothing through God’s omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing. And it would be difficult to find a better illustration of this secret in nature or philosophy; hence I have set on the medallion design IMAGO CREATIONIS [in the image of creation]. It is no less remarkable that there appears therefrom, not only that God made everything from nothing, but also that everything that He made was good; as we can see here, with our own eyes, in this image of creation. Because instead of there appearing no particular order or pattern, as in the common representation of numbers, there appears here in contrast a wonderful order and harmony which cannot be improved upon….
Such harmonious order and beauty can be seen in the small table on the medallion up to 16 or 17; since for a larger table, say to 32, there is not enough room. One can further see that the disorder, which one imagines in the work of God, is but apparent; that if one looks at the matter with the proper perspective, there appears symmetry, which encourages one more and more to love and praise the wisdom, goodness, and beauty of the highest good, from which all goodness and beauty has flowed."
See also Parable.
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Wednesday, March 29, 2006
Wednesday March 29, 2006
Darkness at Noon,
continued
continued
It turns out that Medawar (see previous entry) also wrote a deeply hostile review of Koestler’s The Act of Creation. (See Pluto’s Republic.)
There are plenty more like Medawar, so it may be that a further effort at documentation of Diamond Theory is needed. See this evening’s entry, to follow.
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Tuesday, March 28, 2006
Tuesday March 28, 2006
A Prince of Darkness
“What did he fear? It was not a fear or dread, It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada y pues nada y nada y pues nada. Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee.”
— From Ernest Hemingway,
“A Clean, Well-Lighted Place“
“A Clean, Well-Lighted Place“
| “By groping toward the light we are made to realize how deep the darkness is around us.” — Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118 |
From a review of
Teilhard de Chardin’s
The Phenomenon of Man:
Teilhard de Chardin’s
The Phenomenon of Man:
|
“It would have been |
“He’s good.”
“Good? He’s the fucking
Prince of Darkness!”
— Paul Newman
and Jack Warden
in “The Verdict“
| Sanskrit (transliterated) —
nada: “So Nada Brahma means not only: — Joachim-Ernst Berendt, |
“This book is the outcome of
a course given at Harvard
first by G. W. Mackey….”
— Lynn H. Loomis, 1953, preface to
An Introduction to
Abstract Harmonic Analysis
For more on Mackey and Harvard, see
the Log24 entries of March 14-17.
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Thursday, May 20, 2004
Thursday May 20, 2004
Parable
"A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: 'what is thrown beside' or 'juxtaposed'), a term used to designate the geometric application we call a 'parabola.'…. The basic parables are extended similes or metaphors."
— http://religion.rutgers.edu/nt/
primer/parable.html
"If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind."
— Sharon Begley, "The Puzzle of Genius," Newsweek magazine, June 28, 1993, p. 50
"The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem 'Pied Beauty' has to do with 'creation.' "
— Speaking in Parables, Ch. 2, by Sallie McFague
"The Act of Creation is, I believe, a more truly creative work than any of Koestler's novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls 'bisociation.' And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or 'see,' for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes."
— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
For further details, see
Speaking in Parables:
A Study in Metaphor and Theology
by Sallie McFague
Fortress Press, Philadelphia, 1975
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
"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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