Sunday, November 2, 2025
Monday, August 4, 2025
For the Boys in the Kitchen* —
Yin-Yang Man** at Ninefold Window of Opportunity
Yin-Yang Man** at Ninefold Window of Opportunity
Copenhagen Drafts
Midrash —
Scholium —
|
"… die Schönheit… [ist] die
"Beauty is the proper conformity |
Monday, July 15, 2024
Monday, June 10, 2024
Fast Times for Copenhagen High
Tuesday, March 19, 2024
Metadata: The Copenhagen Interpretation
of “Magic in the Moonlight”
of “Magic in the Moonlight”
Thursday, October 12, 2023
Wednesday, March 13, 2019
The Origin of Change . . .
According to Wallace Stevens:
|
From Savage Logic— Sunday, March 15, 2009 5:24 PM The Origin of Change
A note on the figure
"Two things of opposite natures seem to depend
— Wallace Stevens, |
This post was suggested by the following passage —
" … the Fano plane ,
a set of seven points
grouped into seven lines
that has been called
'the combinatorialist’s coat of arms.' "
— Blake Stacey in a post with tomorrow's date:
… and by Stacey at another weblog, in a post dated Jan. 29, 2019, …
"(Yes, Bohr was the kind of guy who would choose
the yin-yang symbol as his coat of arms.)"
Yes, Stacey is the kind of guy who would casually dismiss
Bohr's coat of arms.
(See also Faust in Copenhagen in this journal)—
» more
Wednesday, March 22, 2017
So Set ’Em Up, Jo
“Danes have been called the happiest people.
I wonder how they measure this.”
— Copenhagen designer in today's online New York Times .
A version of this article is to appear in print on March 26, 2017,
in T Magazine with the headline: "Gray Matters."
See also last night's quarter-to-three post as well as
the webpage "Grids, You Say?" by Norwegian artist Josefine Lyche.
Tuesday, September 29, 2015
Quotes for Michaelmas
A search in this journal for material related to the previous post
on theta characteristics yields…
"The Solomon Key is the working title of an unreleased
novel in progress by American author Dan Brown.
The Solomon Key will be the third book involving the
character of the Harvard professor Robert Langdon,
of which the first two were Angels & Demons (2000) and
The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups,"
by Francesco Dalla Piazza and Bert van Geemen,
May 5, 2008, preprint.
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
"The page numbers
are generally reliable."
For further backstory, click the above link "May 5, 2008,"
which now leads to all posts tagged on080505.
Friday, March 27, 2015
Pursuit of Gestalt*
The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.
Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.
Gestalt-related material from this journal that week —
- Objectivity
- Anatomy of a Cube
- Relativity Problem Revisited
- Symmetric Generation
- Sicilian Reflections
* Title suggested by that of a book by Quine.
Thursday, September 11, 2014
A Class by Itself
The American Mathematical Society yesterday:
Harvey Cohn (1923-2014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."
Thursday, May 30, 2013
Two Characters
Yesterday, May 29, was JFK's birthday.
The above is a belated quotation for that occasion.
Click on the quotation for truth instead of rhetoric.
(Why the above Google Books description of the quotation's
source is in Danish, I do not know. A search for
Bohr + Chinese in this journal may be relevant, as may
also be Faust in Copenhagen.)
Sunday, November 18, 2012
Sermon
Happy birthday to…
Today's sermon, by Marie-Louise von Franz—
For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.
Another modern physicist, Niels Bohr, died
on this date in 1962…
|
The circle above is marked with a version For the square, see the diamond theorem. "Two things of opposite natures seem to depend — Wallace Stevens, |
Thursday, October 11, 2012
Plots
In memory of Copenhagen saxophonist John Tchicai,
who reportedly died last Sunday, October 7, 2012—
Quoted in this journal on the reported date of his death—
But passion ever spins our plots,
And Gretchen is my treasure!
— Wolfgang Pauli character in
Faust in Copenhagen
Related material—
"Physik als Quelle der Spiritualität:
Ein Weg von 'Gretchen' zurück zu 'Beatrice'? "
by Bruno Binggeli
Sunday, October 7, 2012
Faust at Harvard (For Bohr’s Birthday)
Today's Harvard Crimson—
Students Discuss Mental Health
In an effort to break the silence on an often-stigmatized topic,
members of the Harvard community gathered to share
experiences with mental illness through spoken word,
interpretive dance, and candid conversations at Friday night’s
second-annual “Words on the Mind” open mic night.
Related material from this journal on Friday—
The Hallowed Crucible—
Some related symbolism (headings added Oct. 7)—
|
Words:
Applied Mathematics |
The Mind:
Pure Mathematics |
Today's (Sunday, Oct. 7, 2012) Google Doodle for Bohr's birthday—
Review (See also Faust in Copenhagen in this journal)—
» more
Sunday, June 24, 2012
Summer Reading
"What summer reading needs to be is purposeful."
— New York Times opinion column today
One of a set of posts titled "Summer Reading"
in this journal in 2007—
(Click for the original post)
It was only in retrospect
that the silliness
became profound.
— Review of
Faust in Copenhagen
Friday, January 22, 2010
Meta Physics
Friday, September 11, 2009
Friday September 11, 2009
Cover of Underworld,
by Don DeLillo, First Edition,
Advance Reader's Copy, 1997
"Time and chance
happeneth to them all."
— Ecclesiastes 9:11
Related material:
1. The previous entry, on
Copenhagen physicist
Aage Bohr, and
2. Notes from this journal
from Bohr's birthday,
June 19th, through
Midsummer Night, 2007…
including notes on
Faust in Copenhagen
3. Walpurgisnacht 2008 and
Walpurgisnacht 2009
Thursday, July 2, 2009
Thursday July 2, 2009
on a joke by George Carlin,
a passage by Kierkegaard,
and the death on this date
12 years ago
of actor James Stewart
“Thank you, Mr. Twain. Have your people call my people.” –George Carlin on learning he had won the Mark Twain award. Twain’s people were Protestant, Carlin’s Catholic.
The Protestant Kierkegaard:
“… the moment is not properly an atom of time but an atom of eternity. It is the first reflection of eternity in time, its first attempt, as it were, at stopping time….
Once here in Copenhagen there were two actors who probably never thought that their performance could have a deeper significance. They stepped forth onto the stage, placed themselves opposite each other, and then began the mimical representation of one or another passionate conflict. When the mimical act was in full swing and the spectators’ eyes followed the story with expectation of what was to follow, they suddenly stopped and remained motionless as though petrified in the mimical expression of the moment. The effect of this can he exceedingly comical, for the moment in an accidental way becomes commensurable with the eternal.”
Catholic tableau
(with Vivien Leigh
representing the Church)
of Salvation by Works —
Protestant tableau
(with James Stewart
as Protestant Pilgrim)
of Salvation by Grace —
Click on either tableau
for a (much) larger image.
“An unscrupulous jeweler will swap diamonds for cheaper ones when jewelry is dropped off to be sized or repaired, he said.
‘It happens all the time,’ Martin said. ‘Nobody’s watching.'”
Saturday, March 28, 2009
Saturday March 28, 2009
The Rest
of the Story
Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.
| Lotteries on Reba's birthday, 2009 |
Pennsylvania (No revelation) |
New York (Revelation) |
| Mid-day (No belief) |
No belief, no revelation 726 |
Revelation without belief 378 |
| Evening (Belief) |
Belief without revelation 006 |
Belief and revelation 091 |
Interpretations of the evening numbers–
The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.
Clark: Yeah.
Will: Well, I got her number.
How do you like them apples?
The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–
Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–
The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."
marked circle
from March 15:
The circle above is
marked with a version of
the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.
"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."
-- Wallace Stevens,
"Notes Toward a Supreme Fiction,"
Canto IV of "It Must Change"
The square above is marked
with a graphic design
related to the four-diamond
figure of Jung's Aion.
Monday, May 5, 2008
Monday May 5, 2008
"And take upon's
the mystery of things
as if we were God's spies"
— King Lear
From Log24 on Aug. 19, 2003
and on Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::
"Perhaps we are meant to
see the story as a cubist
retelling of the crucifixion."
From Log24 on
Michaelmas 2007:
Google searches suggested by
Sunday's PA lottery numbers
(mid-day 170, evening 144)
and by the above
figure of Kate Beckinsale
pointing to an instance of
the number 144 —
Related material:
Beckinsale in another film
(See At the Crossroads,
Log24, Dec. 8, 2006):
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
From the conclusion of
Joan Didion's 1970 novel
Play It As It Lays —
"I know what 'nothing' means,
and keep on playing."
From Play It As It Lays,
the paperback edition of 1990
(Farrar, Straus and Giroux) —
|
Page 170:
"By the end of a week she was thinking constantly
170
even one micro-second she would have what she had |
"The page numbers
are generally reliable."
Tuesday, April 29, 2008
Tuesday April 29, 2008
at Harvard:
Thomas Wolfe
(Harvard M.A., 1922)
versus
Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)
on
"No culture has a pact with eternity."
— George Steiner, interview in
The Guardian of
"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."
— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005
From today's online Harvard Crimson:
"… under the leadership of Faust,
Harvard students should look forward
to an ever-growing opportunity for
international experience
and artistic endeavor."
Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen
From a recent book
on Wolfgang Pauli,
The Innermost Kernel:
A belated happy birthday
to the late
Felix Christian Klein
(born on April 25) —
Another Harvard figure quoted here on Dec. 5, 2002:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)
From a review of Rosalind Krauss's The Optical Unconscious (MIT Press hardcover, 1993):
Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid-19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."
— Arthur C. Danto in ArtForum, Summer 1993
Rosalind Krauss in The Optical Unconscious (MIT Press paperback, 1994):
For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude Lévi-Strauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.
Wikipedia on the Klein group (denoted V, for Vierergruppe):
In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.
For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:
Monday, April 14, 2008
Monday April 14, 2008
Classical Quantum
From this morning's
New York Times:
… Dr. Wheeler set the agenda for generations of theoretical physicists, using metaphor as effectively as calculus to capture the imaginations of his students and colleagues and to pose questions that would send them, minds blazing, to the barricades to confront nature….
'He rejuvenated general relativity; he made it an experimental subject and took it away from the mathematicians,' said Freeman Dyson, a theorist at the Institute for Advanced Study….
… he [Wheeler] sailed to Copenhagen to work with Bohr, the godfather of the quantum revolution, which had shaken modern science with paradoxical statements about the nature of reality.
'You can talk about people like Buddha, Jesus, Moses, Confucius, but the thing that convinced me that such people existed were the conversations with Bohr,' Dr. Wheeler said….
… Dr. Wheeler was swept up in the Manhattan Project to build an atomic bomb. To his lasting regret, the bomb was not ready in time to change the course of the war in Europe….
Dr. Wheeler continued to do government work after the war, interrupting his research to help develop the hydrogen bomb, promote the building of fallout shelters and support the Vietnam War….
… Dr. Wheeler wondered if this quantum uncertainty somehow applied to the universe and its whole history, whether it was the key to understanding why anything exists at all.
'We are no longer satisfied with insights only into particles, or fields of force, or geometry, or even space and time,' Dr. Wheeler wrote in 1981. 'Today we demand of physics some understanding of existence itself.'
At a 90th birthday celebration in 2003, Dr. Dyson said that Dr. Wheeler was part prosaic calculator, a 'master craftsman,' who decoded nuclear fission, and part poet. 'The poetic Wheeler is a prophet,' he said, 'standing like Moses on the top of Mount Pisgah, looking out over the promised land that his people will one day inherit.'"
— Dennis Overbye, The New York Times,
Monday, April 14, 2008
the poet Wallace Stevens:
"point A / In a perspective
that begins again / At B"
— Wallace Stevens,
"The Rock"
Thursday, November 22, 2007
Thursday November 22, 2007
| A comment at Peter Woit’s weblog today:
T says (3:43 AM today)
My reply (4:26 AM today, awaiting moderation):
|
Related material:
and
Sunday, November 18, 2007
Sunday November 18, 2007
on his birthday, from
the New York Lottery:
Words and Music
Words:
In the Details
"It was only in retrospect
that the silliness became profound.
The players were becoming possessors
of 'a truth with implicit powers
of good and evil,' Gino Segrè writes
in Faust in Copenhagen…
And 'the devil… was in the details.'"
— George Johnson of
The New York Times,
quoted in Log24 on 6/23.
Music:
A Black Berry
"Her wall is filled with pictures,
she gets 'em one by one…."
Chuck Berry, quoted
in Log24 on 2/13.
Related material:
Yesterday's Log24 entry…
Saturday, September 22, 2007
Saturday September 22, 2007
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
Tuesday, July 24, 2007
Tuesday July 24, 2007
See yesterday’s entries for
some relevant quotations
from Wallace Stevens.
Further quotations for what
Marjorie Garber, replying to
a book review by
Frank Kermode, has called
“the Church of St. Frank“–
Frank Kermode on
Harold Bloom:
“He has… a great, almost
selfish passion for poetry,
and he interprets difficult
texts as if there were no
more important activity
in the world, which may
be right.”
Page 348 of Wallace Stevens:
The Poems of Our Climate,
by Harold Bloom
(1977, Cornell U. Press):
… They are more than leaves
that cover the barren rock….
They bear their fruit
so that the year is known….
For more on magic, mysticism, and the Platonic “source of all images,” see Scott McLaren on “Hermeticism and the Metaphysics of Goodness in the Novels of Charles Williams.” McLaren quotes Evelyn Underhill on magic vs. mysticism:
The fundamental difference between the two is this: magic wants to get, mysticism wants to give […] In mysticism the will is united with the emotions in an impassioned desire to transcend the sense-world in order that the self may be joined by love to the one eternal and ultimate Object of love […] In magic, the will unites with the intellect in an impassioned desire for supersensible knowledge. This is the intellectual, aggressive, and scientific temperament trying to extend its field of consciousness […] (Underhill 84; see also 178ff.)
— Underhill, Evelyn. Mysticism: A Study in the Nature and Development of Man’s Spiritual Consciousness. New York: Dutton, 1911.
For more on what Bloom calls the “Will-to-Power over nature,” see Faust in Copenhagen and the recent (20th- and 21st-century) history of Harvard University. These matters are also discussed in “Log24 – Juneteenth through Midsummer Night.”
For more on what Underhill calls “the intellectual, aggressive, and scientific temperament trying to extend its field of consciousness,” see the review, in the August 2007 Notices of the American Mathematical Society, of a book by Douglas Hofstadter– a writer on the nature of consciousness— by magician Martin Gardner.
Saturday, June 23, 2007
Saturday June 23, 2007
A Struggle for
the Soul of Physics
By Gino Segrè
Illustrated. 310 pp.
Viking. $25.95.
“As though their knowledge of the quantum secrets came with the power of prophecy, some three dozen of Europe’s best physicists ended their 1932 meeting in Copenhagen with a parody of Goethe’s ‘Faust.’….
It was only in retrospect that the silliness became profound. The players were becoming possessors of ‘a truth with implicit powers of good and evil,’ Gino Segrè writes in ‘Faust in Copenhagen,’ his inventive new book about the era. And ‘the devil… was in the details.'” –George Johnson
This week’s entries
on Pauli and Faust,
the entries of
June 3 through June 6,
and the five entries
ending on April 7, 2005,
with “In the Details“
Thursday, June 21, 2007
Thursday June 21, 2007
Let No Man
Write My Epigraph
"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."
—Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007
"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."
— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005
"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"
— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)
"… donc Dieu existe, réponse!"
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
— Physics Today
'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"
— The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)
"Pauli linked this symbolism
with the concept of automorphism."
— The Innermost Kernel
(previous entry)
And from
"Symmetry in Mathematics
and Mathematics of Symmetry"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have
The Epigraph–
(Here "whatever" should
of course be "whenever.")
Also from the
Cameron paper:
|
Local or global?
Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:
• exact correspondence of parts; Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them? A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global." |
Some Log24 entries
related to the above politically
(women in mathematics)–
Global and Local:
One Small Step
and mathematically–
Structural Logic continued:
Structure and Logic (4/30/07):
This entry cites
Alice Devillers of Brussels–
"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
as many symmetries as possible."
"There is such a thing
as a tesseract."
Wednesday, June 20, 2007
Wednesday June 20, 2007
Kernel
Mathematical Reviews citation:
MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9
A quote from MR at Amazon.com:
"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
The four-group is also known as the Vierergruppe or Klein group. It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)
The "chessboard" of
Nov. 7, 2006
(as revised Nov. 7, 2012)–
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
Tuesday, June 19, 2007
Tuesday June 19, 2007
Faustus is gone:
regard his hellish fall
— Marlowe
On Faust— today's noon entry and yesterday's "Nightmare Lessons."
On "Meta Physicists"– an entry of June 6, on Cullinane College, has a section titled "Meta Physics."
On Copenhagen— an entry of Bloomsday Eve, 2004 on a native of that city.
"Words, words, words."
— Hamlet
Another metaphysics:
"317 is a prime,
not because we think so,
or because our minds
are shaped in one way
rather than another,
but because it is so,
because mathematical
reality is built that way."
— G. H. Hardy,
A Mathematician's Apology
Saturday, September 17, 2005
Saturday September 17, 2005
continued
From Matt Glaser, Satchmo, the Philosopher:
“… the luminosity and perpetual freshness of Armstrong’s music. These qualities, as well as his essentially abstract ability to affect our perception of time, link him with the other artistic and scientific revolutionaries of the first half of the 20th century. Recently I had a very public fantasy (in Ken Burns’s Jazz) in which Werner Heisenberg attends a Louis Armstrong concert in Copenhagen, in 1933. Did I go too far? Actually, I didn’t go far enough.”
That Log24 entries connect both these dates to Louis Armstrong is, of course, purely coincidental.
Tuesday, September 24, 2002
Tuesday September 24, 2002
The Shining of Lucero
From my journal note, “Shining Forth“:
The Spanish for “Bright Star” is “Lucero.”
The Eye of the Beholder:
When you stand in the dark and look at a star a hundred light years away, not only have the retarded light waves from the star been travelling for a hundred years toward your eyes, but also advanced waves from your eyes have reached a hundred years into the past to encourage the star to shine in your direction.
— John Cramer, “The Quantum Handshake“
From Broken Symmetries, by Paul Preuss, 1983:
He’d toyed with “psi” himself…. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand — for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox….
Peter [Slater] had never thirsted after “hidden variables” to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods.
……………… Those so-called crazy psychics were too sane, that was their problem — they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry. (Ch. 16)
From Secret Passages, by Paul Preuss, 1997:
Minakis caught up and walked beside him in silence, moving with easy strides over the bare ground, listening as Peter [Slater] spoke. “Delos One was ten years ago — quantum theory seemed as natural as water to me then; I could play in it without a care. If I’d had any sense of history, I would have recognized that I’d swallowed the Copenhagen interpretation whole.”
“Back then, you insisted that the quantum world is not a world at all,” Minakis prompted him. “No microworld, only mathematical descriptions.”
“Yes, I was adamant. Those who protested were naive — one has to be willing to tolerate ambiguity, even to be crazy.”
“Bohr’s words?”
“The party line. Of course Bohr did say, ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’ Meaning that when we start to talk what sounds like philosophy, our colleagues should rip us to pieces.” Peter smiled. “They smell my blood already.”
……………… Peter glanced at Minakis. “Let’s say there are indications — I have personal indications — not convincing, perhaps, but suggestive, that the quantum world penetrates the classical world deeply.” He was silent for a moment, then waved his hand at the ruins. “The world of classical physics, I mean. I suppose I’ve come to realize that the world is more than a laboratory.” “We are standing where Apollo was born,” Minakis said. “Leto squatted just there, holding fast to a palm tree, and after nine days of labor gave birth to the god of light and music….”
To Lucero, in memory of
1962 in CuernavacaFrom On Beauty, by Elaine Scarry,
Princeton University Press, 1999 —“Homer sings of the beauty of particular things. Odysseus, washed up on shore, covered with brine, having nearly drowned, comes upon a human community and one person in particular, Nausicaa, whose beauty simply astonishes him. He has never anywhere seen a face so lovely; he has never anywhere seen any thing so lovely….
I have never laid eyes on anyone like you,
neither man nor woman…
I look at you and a sense of wonder takes me.Wait, once I saw the like —
in Delos, beside Apollo’s altar —
the young slip of a palm-tree
springing into the light.”
From Secret Passages, by Paul Preuss, 1997:
“When we try to look inside atoms,” Peter said, “not only can we not see what’s going on, we cannot even construct a coherent picture of what’s going on.”
“If you will forgive me, Peter,” Minakis said, turning to the others. “He means that we can construct several pictures — that light and matter are waves, for example, or that light and matter are particles — but that all these pictures are inadequate. What’s left to us is the bare mathematics of quantum theory.”
…. “Whatever the really real world is like, my friend, it is not what you might imagine.”
……………… Talking physics, Peter tended to bluntness. “Tell me more about this real world you imagine but can’t describe.”Minakis turned away from the view of the sunset. “Are you familiar with John Cramer’s transactional interpretation of quantum mechanics?”
“No I’m not.”
……………… “Read Cramer. I’ll give you his papers. Then we can talk.”
From John Cramer, “The Quantum Handshake“:
Advanced waves could perhaps, under the right circumstances, lead to “ansible-type” FTL communication favored by Le Guin and Card….
For more on Le Guin and Card, see my journal notes below.
For more on the meaning of “lucero,” see the Wallace Stevens poem “Martial Cadenza.”