For posts related to the title, click here.
Friday, December 27, 2013
Saturday, January 14, 2017
1984: A Space Odyssey
See Eightfold 1984 in this journal.
Related material —
"… the object sets up a kind of
frame or space or field
within which there can be epiphany."
"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989
"Perhaps every science must start with metaphor
and end with algebra; and perhaps without the metaphor
there would never have been any algebra."
— Max Black, Models and Metaphors ,
Cornell University Press, Ithaca, NY, 1962
Click to enlarge:
Tuesday, January 10, 2017
Eightfold Epiphany
The reported death today at 105 of an admirable war correspondent,
"a perennial fixture at the Foreign Correspondents’ Club in Hong Kong,"
suggested a search in this journal for that city.
The search recalled to mind a notable quotation from
a Montreal philosopher —
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)
For some context, see St. Lucia's Day, 2012.
See also Epiphany 2017 —
Thursday, July 23, 2015
Space
It's Space Week at Camp Google.
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
— Charles Taylor
“My little baby sister can do it with ease.
It’s easier to learn than those ABC’s.”
— Kylie Minogue
Friday, February 14, 2014
Epiphany
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
Related material —
Star Wars (January 11, 2014),
The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —
— and the March 2014 issue of the
Notices of the American Mathematical Society —
See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices on the Shaw prize —
"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong."
Thursday, December 13, 2012
Space
… The sequel to Vibrations
Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
Or place.
See A Prince of Darkness
and "A Clean, Well-Lighted Place."
Thursday, January 6, 2011
Epiphany Riddle
"Spaces and geometries, those which we perceive,
which we can’t perceive, or which only some of us perceive,
are a recurring theme in Against the Day ."
"大哉大哉 宇宙之谜
美哉美哉 真理之源"
"Great indeed is the riddle of the universe.
Beautiful indeed is the source of truth."
— Shing-Tung Yau, Chairman,
Department of Mathematics, Harvard University
"Always keep a diamond in your mind."
— King Solomon at the Paradiso
Image from stoneship.org
Friday, August 25, 2023
On the Night Road from Marfa
"I’m really interested in exploring space."
— New Yorker cover artist for the Aug. 28, 2023, issue.
Related cinematic art . . .
From a search in this journal for Nocturnal —
For some Bright Art Blocks Moments , see Cube Epiphany .
Sunday, October 11, 2020
Saniga on Einstein
See “Einstein on Acid” by Stephen Battersby
(New Scientist , Vol. 180, issue 2426 — 20 Dec. 2003, 40-43).
That 2003 article is about some speculations of Metod Saniga.
“Saniga is not a professional mystic or
a peddler of drugs, he is an astrophysicist
at the Slovak Academy of Sciences in Bratislava.
It seems unlikely that studying stars led him to
such a way-out view of space and time. Has he
undergone a drug-induced epiphany, or a period
of mental instability? ‘No, no, no,’ Saniga says,
‘I am a perfectly sane person.'”
Some more recent and much less speculative remarks by Saniga
are related to the Klein correspondence —
arXiv.org > math > arXiv:1409.5691:
Mathematics > Combinatorics
[Submitted on 17 Sep 2014]
The Complement of Binary Klein Quadric
as a Combinatorial Grassmannian
By Metod Saniga
“Given a hyperbolic quadric of PG(5,2), there are 28 points
off this quadric and 56 lines skew to it. It is shown that the
(286,563)-configuration formed by these points and lines
is isomorphic to the combinatorial Grassmannian of type
G2(8). It is also pointed out that a set of seven points of
G2(8) whose labels share a mark corresponds to a
Conwell heptad of PG(5,2). Gradual removal of Conwell
heptads from the (286,563)-configuration yields a nested
sequence of binomial configurations identical with part of
that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).”
Related entertainment —
See Log24 on the date, 17 Sept. 2014, of Saniga’s Klein-quadric article:
Friday, May 15, 2020
Review
Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
See also Talking of Michelangelo.
Related material for comedians —
Literature ad absurdum —
Tuesday, July 9, 2019
Dreamtimes
“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”
— John Malkovich at squarespace.com, January 10, 2017
Also on that date . . .
Wednesday, April 12, 2017
Knucklebone
"The state of the universe, physicists say, is a cosmological
relic—a glass ark with hammered-gold seams, pip trapped inside, god’s
knucklebone, nanosecond high-energy outward burst—kaboom!—
and space fills up with proto-stars . . . ."
— "A Natural History of Light," a poem by Marsha de la O,
The New Yorker , issue of December 12, 2016
"Angel's Bone," by Du Yun, premiered on January 6, 2016 . . . ."
Saturday, August 30, 2014
Physics and Theology
The titles of the previous three posts refer to
Hermann Weyl’s 1918 book Raum, Zeit, Materie
(Space, Time, Matter).
This suggests a look at a poetically parallel 1950 title —
The Lion, the Witch and the Wardrobe —
and at its underlying philosophy:
I am among “those who do not know that this great myth became Fact.”
I do, however, note that some other odd things have become fact.
Those who wish more on this topic may consult:
- The Metaphysics of the Incarnation , from
Oxford University Press on Epiphany 2011. - Epiphany Riddle, from that same date.
- Incarnation in this journal.
Thursday, February 13, 2014
Plan 9
The final link in today's previous post leads to
a post whose own final link leads to…
Thursday, December 13, 2012
|
Wednesday, December 12, 2012
Chromatic Plenitude
(Continued from 2 PM ET Tuesday)
“… the object sets up a kind of frame or space or field
within which there can be epiphany.”
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)
"The absolute consonance is a state of chromatic plenitude."
"… the nearest precedent might be found in Becky Sharp .
The opening of the Duchess of Richmond's ball,
with its organization of strong contrasts and
display of chromatic plenitude, presents a schema…."
— Scott Higgins, Harnessing the Technicolor Rainbow:
Color Design in The 1930s , University of Texas Press,
2007, page 142
Note the pattern on the dance floor.
(Click for wider image.)
"At the still point…" — Four Quartets
Monday, June 29, 2009
Monday June 29, 2009
for St. Peter's Day
"Have your people
call my people."
— George Carlin
Diamond life, lover boy;
we move in space
with minimum waste
and maximum joy.
— Sade, quoted here on
Lincoln's Birthday, 2003
This is perhaps suitable
for the soundtrack of
the film "Blockheads"
(currently in development)–
Diamond Life —
Related material from Wikipedia:
"Uta Frith, in her book Autism: Explaining the Enigma,[5] addresses the superior performance of autistic individuals on the block design [link not in Wikipedia] test. This was also addressed in [an] earlier paper.[6] A particularly interesting article demonstrates the differences in construction time in the performance of the block design task by Asperger syndrome individuals and non-Asperger's individuals. An essential point here is that in an unsegmented version of the task, Asperger's individuals performed dramatically faster than non-Asperger's individuals: [7]."
5. Frith, Uta (2003). Autism: explaining the enigma (2nd ed. ). Cambridge, MA: Blackwell Pub. ISBN 0-631-22901-9.
6. Shah A, Frith U (Nov 1993). "Why do autistic individuals show superior performance on the block design task?". J Child Psychol Psychiatry 34 (8): 1351–64. PMID 8294523.
7. Caron MJ, Mottron L, Berthiaume C, Dawson M (Jul 2006). "Cognitive mechanisms, specificity and neural underpinnings of visuospatial peaks in autism". Brain 129 (Pt 7): 1789–802. doi: . PMID 16597652. "Fig 3".
Related material from a film (see Calvinist Epiphany, June 17):
For the relevance of this maxim to autism, see Markoff Process (March 4, 2009).
Friday, May 22, 2009
Friday May 22, 2009
New York Times
banner this morning:
Related material from
July 11, 2008:
The HSBC Logo Designer — Henry Steiner He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design. Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club. His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995). |
Charles Taylor,
"Epiphanies of Modernism," Chapter 24 of Sources of the Self (Cambridge U. Press, 1989, p. 477):
"… the object sets up
See also Talking of Michelangelo.
|
Related material suggested by
an ad last night on
ABC's Ugly Betty season finale:
Diamond from last night's
Log24 entry, with
four colored pencils from
Diane Robertson Design:
See also
A Four-Color Theorem.
Saturday, April 25, 2009
Saturday April 25, 2009
Hotel Puzzle by John Tierney "Russell Crowe arrives at the Hotel Infinity looking tired and ornery. He demands a room. The clerk informs him that there are no vacancies…."
|
Footprints from California today
(all by a person or persons using Firefox browsers):
7:10 AM
http://m759.xanga.com/679142359/concepts-of-space/?
Concepts of Space: Euclid vs. Galois
8:51 AM
http://m759.xanga.com/689601851/art-wars-continued/?
Art Wars continued: Behind the Picture
1:33 PM
http://m759.xanga.com/678995132/a-riff-for-dave/?
A Riff for Dave: Me and My Shadow
2:11 PM
http://m759.xanga.com/638308002/a-death-of-kings/?
A Death of Kings: In Memory of Bobby Fischer
2:48 PM
http://m759.xanga.com/691644175/art-wars-in-review–/?
Art Wars in review– Through the Looking Glass: A Sort of Eternity
3:28 PM and
http://m759.xanga.com/684680406/annals-of-philosophy/?
Annals of Philosophy: The Dormouse of Perception
4:28 PM
http://m759.xanga.com/641536988/epiphany-for-roy-part-i/?
Epiphany for Roy, Part I
6:03 PM
http://m759.xanga.com/641949564/art-wars-continued/?
At the Still Point: All That Jazz
6:22 PM
http://m759.xanga.com/644330798/where-entertainment-is-not-god/?
Where Entertainment is Not God: The Just Word
7:14 PM
http://m759.xanga.com/643490468/happy-new-yorker-day/?
Happy New Yorker Day– Class Galore
7:16 PM
http://m759.xanga.com/643812753/the-politics-of-change/?
The Politics of Change: Jumpers
"We are programmed to receive."
— Hotel California
Tuesday, January 6, 2009
Tuesday January 6, 2009
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson's 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung's theory of archetypes:
"… we do not need to accept Jung’s theory as true in order to find it illuminating."
The same is true of Jung's remarks on synchronicity.
For example —
Yesterday's entry, "A Wealth of Algebraic Structure," lists two articles– each, as it happens, related to Jung's four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis's 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis's 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 1984-2003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
"What, then, is a frame, and what is frame work?"
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 "frame of reference" found in the above Cambridge University Press articles.
windows of knowledge."
— Vladimir Nabokov
Friday, July 11, 2008
Friday July 11, 2008
The HSBC Logo Designer — Henry Steiner He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design. Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club. His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995). |
from the past —
Charles Taylor,
"… the object sets up
See also Talking of Michelangelo.
|
Related material
from today —
Escape from a
cartoon graveyard:
Thursday, June 26, 2008
Thursday June 26, 2008
Having survived that ominous date, I feel it is fitting to review what Wallace Stevens called "Credences of Summer"– religious principles for those who feel that faith and doubt are best reconciled by art.
"Credences of Summer," VII,
by Wallace Stevens, from
"Three times the concentred |
Definition of Epiphany
From James Joyce's Stephen Hero, first published posthumously in 1944. The excerpt below is from a version edited by John J. Slocum and Herbert Cahoon (New York: New Directions Press, 1959).
Three Times: … By an epiphany he meant a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance: — Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin's street furniture. Then all at once I see it and I know at once what it is: epiphany. — What? — Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty. — Yes? said Cranly absently. — No esthetic theory, pursued Stephen relentlessly, is of any value which investigates with the aid of the lantern of tradition. What we symbolise in black the Chinaman may symbolise in yellow: each has his own tradition. Greek beauty laughs at Coptic beauty and the American Indian derides them both. It is almost impossible to reconcile all tradition whereas it is by no means impossible to find the justification of every form of beauty which has ever been adored on the earth by an examination into the mechanism of esthetic apprehension whether it be dressed in red, white, yellow or black. We have no reason for thinking that the Chinaman has a different system of digestion from that which we have though our diets are quite dissimilar. The apprehensive faculty must be scrutinised in action. — Yes … — You know what Aquinas says: The three things requisite for beauty are, integrity, a wholeness, symmetry and radiance. Some day I will expand that sentence into a treatise. Consider the performance of your own mind when confronted with any object, hypothetically beautiful. Your mind to apprehend that object divides the entire universe into two parts, the object, and the void which is not the object. To apprehend it you must lift it away from everything else: and then you perceive that it is one integral thing, that is a thing. You recognise its integrity. Isn't that so? — And then? — That is the first quality of beauty: it is declared in a simple sudden synthesis of the faculty which apprehends. What then? Analysis then. The mind considers the object in whole and in part, in relation to itself and to other objects, examines the balance of its parts, contemplates the form of the object, traverses every cranny of the structure. So the mind receives the impression of the symmetry of the object. The mind recognises that the object is in the strict sense of the word, a thing, a definitely constituted entity. You see? — Let us turn back, said Cranly. They had reached the corner of Grafton St and as the footpath was overcrowded they turned back northwards. Cranly had an inclination to watch the antics of a drunkard who had been ejected from a bar in Suffolk St but Stephen took his arm summarily and led him away. — Now for the third quality. For a long time I couldn't make out what Aquinas meant. He uses a figurative word (a very unusual thing for him) but I have solved it. Claritas is quidditas. After the analysis which discovers the second quality the mind makes the only logically possible synthesis and discovers the third quality. This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany. Having finished his argument Stephen walked on in silence. He felt Cranly's hostility and he accused himself of having cheapened the eternal images of beauty. For the first time, too, he felt slightly awkward in his friend's company and to restore a mood of flippant familiarity he glanced up at the clock of the Ballast Office and smiled: — It has not epiphanised yet, he said. |
Under the Volcano,
by Malcolm Lowry, "What have I got out of my life? Contacts with famous men… The occasion Einstein asked me the time, for instance. That summer evening…. smiles when I say I don't know. And yet asked me. Yes: the great Jew, who has upset the whole world's notions of time and space, once leaned down… to ask me… ragged freshman… at the first approach of the evening star, the time. And smiled again when I pointed out the clock neither of us had noticed."
An approach of
This figure is from a webpage,
As noted in yesterday's early- "The appearance of the evening star brings with it long-standing notions of safety within and danger without. In a letter to Harriet Monroe, written December 23, 1926, Stevens refers to the Sapphic fragment that invokes the genius of evening: 'Evening star that bringest back all that lightsome Dawn hath scattered afar, thou bringest the sheep, thou bringest the goat, thou bringest the child home to the mother.' Christmas, writes Stevens, 'is like Sappho's evening: it brings us all home to the fold' (Letters of Wallace Stevens, 248)."
— Barbara Fisher, |
Wednesday, June 25, 2008
Wednesday June 25, 2008
“I would not know what the spirit
of a philosopher might wish more
to be than a good dancer.
For the dance is his ideal,
also his art, and finally also his
only piety, his ‘service of God.'”
Charles Taylor, winner
of this year’s Kyoto Prize
in arts and philosophy:
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
“My little baby sister
can do it with ease.
It’s easier to learn
than those ABC’s.”
Wednesday June 25, 2008
Other approaches to the
eight-ray star figure
have been sketched in
various Log24 entries.
See, for instance, the
June 21 entries on
the Kyoto Prize for
arts and philosophy.
Quine won this prize
in 1996.
Quine’s figure, cited in an
argument against universals,
is also a classic symbol for
the morning or evening star.
This year’s winner
of the Kyoto Prize has
a more poetic approach
to philosophy:
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
For one such frame or space,
a Mexican cantina, see
Shining Forth.
See also Damnation Morning and
The Devil and Wallace Stevens.
Charles Taylor. See
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)
Saturday, June 21, 2008
Saturday June 21, 2008
for lifetime achievement
in arts and philosophy
this year goes to
Charles Taylor,
Montreal philosophy professor.
“The Kyoto Prize has been given in three domains since 1984:
advanced technology, basic sciences, and the arts and philosophy.
It is administered by the Inamori Foundation, whose president,
Kazuo Inamori, is founder and chairman emeritus of Kyocera and
KDDI Corporation, two Japanese telecommunications giants.”
“The Kyocera brand symbol is composed of a corporate mark |
Related material —
Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
See also Talking of Michelangelo.
Thursday, May 22, 2008
Thursday May 22, 2008
An Exercise in
Conceptual Art
THE JUDGMENT
Undertakings bring misfortune.
Nothing that would further.
“Brian O’Doherty, an Irish-born artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
— New York Times, May 22, 2008
THE IMAGE
Thus the superior man
understands the transitory
in the light of
the eternity of the end.
Another version of
the image:
See 2/22/08
and 4/19/08.
Michael Kimmelman in today’s New York Times—
“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”
Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20th-century art:
“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. We have now reached a point where we see not the art but the space first…. An image comes to mind of a white, ideal space that, more than any single picture, may be the archetypal image of 20th-century art.”
“Nothing that would further.”
— Hexagram 54
…. Now thou art an 0 |
“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead.”
— The Greater Trumps,
by Charles Williams, Ch. 14
Monday, April 28, 2008
Monday April 28, 2008
The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.
One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.
The following
figure does
allow such
an epiphany.
One approach to
the epiphany:
"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein
The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:
For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
Saturday, April 28, 2007
Saturday April 28, 2007
See last year’s
entries for 5/10 —
My Space
and for 2/23 —
Cubist Epiphany
“This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”
— Robert A. Heinlein,
The Number of the Beast
Tuesday, January 9, 2007
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, |
Thursday, February 23, 2006
Thursday February 23, 2006
“In The Painted Word, a rumination on the state of American painting in the 1970s, Tom Wolfe described an epiphany….”
— Peter Berkowitz, “Literature in Theory”
“I had an epiphany.”
— Apostolos Doxiadis, organizer of last summer’s conference on mathematics and narrative. See the Log24 entry of 1:06 PM last August 23 and the four entries that preceded it.
“… das Durchleuchten des ewigen Glanzes des ‘Einen’ durch die materielle Erscheinung“
— A definition of beauty from Plotinus, via Werner Heisenberg
“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, quoted in The Shining of May 29
“Perhaps we are meant to see the story as a cubist retelling of the crucifixion….”
— Adam White Scoville, quoted in Cubist Crucifixion, on Iain Pears’s novel, An Instance of the Fingerpost
Tuesday, November 29, 2005
Tuesday November 29, 2005
“The Chinese… speak of a great thing (the greatest thing) called the Tao. It is the reality beyond all predicates, the abyss that was before the Creator Himself. It is Nature, it is the Way, the Road. It is the Way in which the universe goes on, the Way in which things everlastingly emerge, stilly and tranquilly, into space and time. It is also the Way which every man should tread in imitation of that cosmic and supercosmic progression, conforming all activities to that great exemplar.”
— C. S. Lewis in The Abolition of Man
“In his preface to That Hideous Strength, Lewis says the novel has a serious point that he has tried to make in this little book, The Abolition of Man. The novel is a work of fantasy or science fiction, while Abolition is a short philosophical work about moral education, but as we shall see the two go together; we will understand either book better by having read and thought about the other.”
— Dale Nelson, Notes on The Abolition of Man
“In Epiphany Term, 1942, C.S. Lewis delivered the Riddell Memorial Lectures… in…. the University of Durham…. He delivered three lectures entitled ‘Men without Chests,’ ‘The Way,’ and ‘The Abolition of Man.’ In them he set out to attack and confute what he saw as the errors of his age. He started by quoting some fashionable lunacy from an educationalists’ textbook, from which he developed a general attack on moral subjectivism. In his second lecture he argued against various contemporary isms, which purported to replace traditional objective morality. His final lecture, ‘The Abolition of Man,’ which also provided the title of the book published the following year, was a sustained attack on hard-line scientific anti-humanism. The intervening fifty years have largely vindicated Lewis.”
— J. R. Lucas, The Restoration of Man
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, January 6, 2004
Tuesday January 6, 2004
720 in the Book
Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.
An article titled On Mathematical Imagination concludes by looking forward to
“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”
Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.
Hmm.
The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:
Abel’s Proof: An Essay
on the Sources and Meaning
of Mathematical Unsolvability
by Peter Pesic,
MIT Press, 2003
From a review:
“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….
Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations. The reader is left with little clarity on this sequel to the story….”
— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244
Here, it seems, is my epiphany:
“Elliptic modular functions suffice to solve all polynomial equations.”
Incidental Remarks
on Synchronicity,
Part I
Those who seek a star
on this Feast of the Epiphany
may click here.
Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.
Just how such equations can be solved is a less familiar story. I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions. Modular functions are also distantly related, via the topic of “moonshine” and via the “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.
Incidental Remarks
on Synchronicity,
Part II
There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.
Here is what I was able to find on the Web about Pesic’s claim:
From Wolfram Research:
From Solving the Quintic —
“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”
From Siegel Theta Function —
“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”
From Polynomial —
“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”
Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.
Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999.
From Angel Zhivkov,
Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:
“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker: in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function…. Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist. This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions. In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”
— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants“
Incidental Remarks
on Synchronicity,
Part III
From Music for Dunne’s Wake:
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
“720 in |
“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.
Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.
For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see
For the relevance of the time
of this entry, 10:10, see
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Related recreational reading:
Labyrinth |
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