See also Icon and a fresh New York Times obituary.
Thursday, March 19, 2015
Midnight in the Garden continues…
Friday, March 13, 2015
Mathieu Moonshine
(Continued from yesterday's "earlier references" link.)
Yesterday at the Simons Foundation's Quanta Magazine :
See also earlier Log24 references to Mathieu moonshine .
I do not know the origin of this succinct phrase, taken from
an undated web page of Anne Taormina.
Friday, January 30, 2015
The Relaxed Field*
In memory of a dead poet —
"Relax," said the night man.
"We are programmed to receive."
* A phrase from a new book by mathematician
Michael Harris, Mathematics without Apologies .
Monday, December 15, 2014
May We?
John Burt Foster Jr. in Nabokov’s Art of Memory and
European Modernism (Princeton U. Press, 1993, p. 224) —
At the time of The Waste Land , in a comment on
Joyce’s Ulysses that influenced many later definitions
of modernism in the English-speaking world, Eliot
announced, “instead of narrative method, we may
now use the mythical method.”13
May we? … Further details —
From T. S. Eliot, “‘Ulysses,’ Order and Myth,” the last two paragraphs: It is here that Mr Joyce’s parallel use of the Odyssey has a great importance. It has the importance of a scientific discovery. No one else has built a novel upon such a foundation before: it has never before been necessary. I am not begging the question in calling Ulysses a novel; and if you call it an epic it will not matter. If it is not a novel, that is simply because the novel is a form which will no longer serve; it is because the novel, instead of being a form, was simply the expression of an age which had not sufficiently lost all form to feel the need of something stricter. Mr Joyce has written one novel – the Portrait ; Mr Wyndham Lewis has written one novel – Tarr . I do not suppose that either of them will ever write another “novel.” The novel ended with Flaubert and with James. It is, I think, because Mr Joyce and Mr Lewis, being “in advance” of their time, felt a conscious or probably unconscious dissatisfaction with the form, that their novels are more formless than those of a dozen clever writers who are unaware of its obsolescence. In using the myth, in manipulating a continuous parallel between contemporaneity and antiquity, Mr Joyce is pursuing a method which others must pursue after him. They will not be imitators, any more than the scientist who uses the discoveries of an Einstein in pursuing his own, independent, further investigations. It is simply a way of controlling, of ordering, of giving a shape and a significance to the immense panorama of futility and anarchy which is contemporary history. It is a method already adumbrated by Mr Yeats, and of the need for which I believe Mr Yeats to have been the first contemporary to be conscious. It is a method for which the horoscope is auspicious. Psychology (such as it is, and whether our reaction to it be comic or serious), ethnology, and <i”>The Golden Bough have concurred to make possible what was impossible even a few years ago. Instead of narrative method, we may now use the mythical method. It is, I seriously believe, a step towards making the modern world possible for art, toward that order and form which Mr Aldington so earnestly desires. And only those who have won their own discipline in secret and without aid, in a world which offers very little assistance to that end, can be of any use in furthering this advance. </i”> |
Monday, December 1, 2014
Change Arises
Flashback to St. Andrew's Day, 2013 —
Saturday, November 30, 2013
|
If the object is a cube, change arises from the fact
that the object has six faces…
and is the unit cell for the six -dimensional
hyperspace H over the two-element field —
A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —
Wednesday, November 5, 2014
Dark Fields…
From the first of previous Log24 posts tagged “Dark Fields”—
“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:
On the Graphs of Hoffman-Singleton and Higman-Sims.
His truth is marching on.”
See also Foundation Square (October 25, 2014).
Sunday, August 10, 2014
Sermon
From The Mathematics of Language:
10th and 11th Biennial Conference….
Berlin, Springer, 2010 —
“Creation Myths of Generative Grammar
and the Mathematics of Syntactic Structures”
by Geoffrey K. Pullum, University of Edinburgh
Abstract
“Syntactic Structures (Chomsky [6]) is widely believed to have laid
the foundations of a cognitive revolution in linguistic science, and
to have presented (i) the first use in linguistics of powerful new ideas
regarding grammars as generative systems, (ii) a proof that English
was not a regular language, (iii) decisive syntactic arguments against
context-free phrase structure grammar description, and (iv) a
demonstration of how transformational rules could provide a formal
solution to those problems. None of these things are true. This paper
offers a retrospective analysis and evaluation.”
Sunday, July 20, 2014
Saturday, July 19, 2014
Zero Art
“Zero is a group of artists founded by Heinz Mack and
Otto Piene….” — Wikipedia
“The title ZERO was the result of months of search
and was finally found more or less by chance.
From the beginning we looked upon the term
not as an expression of nihilism – or a dada-like gag,
but as a word indicating a zone of silence and of
pure possibilities for a new beginning as at the
count-down when rockets take off- zero is the
incommensurable zone in which the old state turns
into the new.”
Alternate Reality
(Where Entertainment Is God , continued)
In memory of artist Otto Piene — a news item from last May
at the ZERO Foundation website on an exhibition that closes tomorrow —
2014-05-15
Today is the opening of the exhibition ZERO — Zwischen Himmel und Erde
in Friedrichshafen. The Zeppelin Museum is showing wonderful artworks
all related to heaven and earth by various ZERO artists
such as Piene, Mack, Uecker, Klein, Luther, and Manzoni.
ZERO – Zwischen Himmel und Erde
Zeppelin Museum Friedrichshafen
15.05. – 20.07.2014
www.zeppelin-museum.de
“Oh, show me the way to the next whiskey bar”
— Song lyric from previous post
“In a technologically advanced 1939, the zeppelin Hindenburg III
arrives in New York City, mooring atop the Empire State Building.”
— Wikipedia on the first scene of the 2004 film
“Sky Captain and the World of Tomorrow“
Friday, July 4, 2014
Knockout
(Continued from yesterday’s noon post, from “Block That Metaphor,”
and from “Mystery Box III: Inside, Outside“)
“In one corner are the advocates of the Common Core,
led by the Bill and Melinda Gates Foundation, which
helped develop the standards and has defended them
against efforts by some states to roll them back. In the
challengers’ corner, a lineup of foundations and
philanthropists…. Other funders in the opponents’ corner
read like a ‘who’s who’ of well-heeled conservative
philanthropists, including Pittsburgh media magnate
Richard Mellon Scaife….”
— “Meet the Funders Fighting the Common Core,”
from Inside Philanthropy , Feb. 10, 2014
Scaife reportedly died this morning.
Thursday, July 3, 2014
Gates and Windows:
“Gates said his foundation is an advocate for the Common Core State Standards
that are part of the national curriculum and focus on mathematics and language
arts. He said learning ‘needs to be on the edge’ where it is challenging but not
too challenging, and that students receive the basics through Common Core.
‘It’s great to teach other things, but you need that foundation,’ he said.”
— T. S. Last in the Albuquerque Journal , 12:05 AM Tuesday, July 1, 2014
See also the previous post (Core Mathematics: Arrays) and, elsewhere
in this journal,
“Eight is a Gate.” — Mnemonic rhyme:
Thursday, May 1, 2014
Thursday, April 10, 2014
Change Arises (continued)
In memory of Lucia Eames, who reportedly died
on April 1, 2014:
“… Walter Gropius was her professor ….”
See also in this journal Gropius and the April 1 posts.
Related material: “As a little child.”
Wednesday, April 2, 2014
Change Arises
Wednesday, March 26, 2014
Sinai Prize
Yakov G. Sinai today won the 2014 Abel Prize.
Earlier, he won the Wolf Prize.
Wolf Foundation press release quoted in the March 1997
Notices of the American Mathematical Society —
On Sinai —
“He is generally recognized as the world leader
in the mathematics of statistical physics.”
This afternoon’s New York Lottery: 813 and 1857.
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."
Wednesday, January 15, 2014
Entertainment Theory
From "Entertainment," a 1981 story by M. A. Foster—
"For some time, Cormen had enjoyed a peculiar suspicion, which he had learned from his wanderings around the city, and cultivated with a little notebook, in which he had made a detailed series of notes and jottings, as well as crude, but effective, charts and maps of certain districts. 'Cormen's Problem,' as it was known, was familiar to the members of the circle in which he moved; in fact, if he had not been so effective with his productions and so engaging in his personality, they might have considered him a bore. It seemed, so the suspicion went, that the city was slowly shrinking, as evidenced by abandoned districts along the city edges. Beyond the empty houses were ruins, and beyond that, traces of foundations and street lines. Moreover, it had recently dawned on him that there were no roads out of the city, although there were no restraints. One hardly noticed this—it was the norm. But like many an easy assumption, once broken it became increasingly obvious. Cormen's acquaintances were tolerant of his aberration, but generally unsympathetic. What he needed was proof, something he could demonstrate in black and white—and color if required. But the city was reluctant, so it appeared, to give up its realities so easily. The Master Entertainment Center, MEC, would not answer direct queries about this, even though it would obediently show him presentations, pictorial or symbolic as he required, of the areas in question. But it was tiresome detail work, in which he had to proceed completely on his own." |
Lily Collins in City of Bones (2013)—
American Folk Art (see August 23, 2011) —
See as well Ballet Blanc .
Thursday, December 12, 2013
Communion of Saints
Saturday, December 7, 2013
Shadow of a Lie
A recent addition to Barry Mazur's home page—
"December 1, 2013: Here are rough notes for
a short talk entitled The Faces of Evidence
(in Mathematics) ([PDF]) to be given at the
Cambridge Scientific Club, Dec. 5 2013."
The PDF link does not work, but some earlier remarks by
Mazur on this topic have been published elsewhere:
Related material:
The Proof and the Lie (St. Andrew's Day, 2003), and
a recent repetition of the lie in Wikipedia:
"Around 1955, Japanese mathematicians Goro Shimura
and Yutaka Taniyama observed a possible link between
two apparently completely distinct, branches of mathematics,
elliptic curves and modular forms."
This statement, from the article on Algebraic number theory,
was added on Oct. 22, 2013 by one "Brirush," apparently a
Temple University postdoctoral researcher, in what he rightly
called a "terrible history summary."
Saturday, November 30, 2013
Waiting for Ogdoad
Continued from October 30 (Devil’s Night), 2013.
“In a sense, we would see that change
arises from the structure of the object.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013
This suggests a review of mathematics and the
“Classic of Change ,” the I Ching .
The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching as the corners of a cube.
See also…
(Click for clearer image.)
The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching trigrams and as
seven natural ways of slicing the cube.
For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube in a post of April 9, 2012.
Friday, November 29, 2013
Odd Facts
"These are odd facts…." — G. H. Hardy,
quoted in the previous post, "Centered"
Other odd facts:
If n is odd, then the object at the center
of the n×n square is a square.
Similarly for the n×n×n cube.
Related meditation:
“In a sense, we would see that change
arises from the structure of the object,” he said.
“But it’s not from the object changing.
The object is basically timeless.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013,
"A Jewel at the Heart of Quantum Physics"
See also "My God, it's full of… everything."
Monday, November 25, 2013
Windows
Ben Brantley reviewing a show by the X-Men patriarchs
that opened on Sunday:
"This isn’t just a matter of theatergoers chuckling
to show that they’re smart and cultured and had
damn well better be having a good time after
forking out all that money…."
I prefer reality (which includes the life of Fred Kavli) :
See also Saturday's posts Chess and Frame Tale.
Whether the patriarch Kavli, pictured above, is now having
a good time, I do not know. I hope so.
Saturday, November 23, 2013
Chess
Norwegian, 22, Takes World Chess Title
Quoted here on Thursday, the date of Kavli‘s death:
Herbert Mitgang’s New York Times
obituary of Cleanth Brooks—
“The New Critics advocated close reading of literary texts
and detailed analysis, concentrating on semantics, meter,
imagery, metaphor and symbol as well as references to
history, biography and cultural background.”
See also Steiner, Chess, and Death.
Monday, October 7, 2013
The Book of Abraham
On Abraham Nemeth, the developer of Braille
for blind students of mathematics —
"… he began tinkering with the six-dot cell
that is the foundation of Braille."
For a different six-dot cell, see Nocciolo .
"Throughout his life, he dedicated much of his
spare time to creating Braille versions of Jewish
texts, including helping to proofread a Braille
Hebrew Bible in the 1950s." — Nemeth's obituary
Those who prefer entertainment may consult The Book of Eli.
Friday, September 6, 2013
Space
"A vast space that travels down to the bedrock
upon which the towers were built, the museum
winds its way deeper and deeper underground,
taking visitors on a journey to the very bottom."
— The Associated Press in
this evening's Washington Post
This suggests a review of a different sort of
bedrock:—
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Sunday, July 21, 2013
Comic-Con
This is the weekend for Comic-Con International in San Diego.
The convention includes an art show. (Click above image to enlarge.)
Related material from Norway…
Suggested nominations for a Kavli Prize:
1. Josefine Lyche's highly imaginative catalog page for
the current Norwegian art exhibition I de lange nætter,
which mentions her interest in sacred geometry
2. Sacred Geometry: Drawing a Metatron Cube
… and from San Diego—
The Kavli Institutes logo:
Saturday, July 20, 2013
Vril Chick
A Brainstorm for Jo Lyxe :
Swastika logo of the Kavli Institute for Brain and Mind
at the University of California, San Diego.
Soundtrack: Under the Iron Sky .
Thursday, June 20, 2013
ART WARS: Chesterton Thursday
The New York Times philosophy column "The Stone"
last evening had an essay on art by a sarcastic anarchist,
one Crispin Sartwell—
"… whole generations of art lovers have been
trained in modernist dogma, and arts institutions’
access to various forms of state or foundation
support depend on it completely. One goes to
the museum to gasp at stunning works of
incomparable, super-human genius by beings
who are infinitely more exalted and important
than the mere humans staring at their paintings.
That’s why ordinary people staring at a Picasso
(allegedly) experience a kind of transcendence
or re-articulation of their lives and world."
Cubism Re-Articulated:
Click image for some backstory.
(IMAGE: Walter Gropius and Froebel's Third Gift,
from a Google image search today)
Background: Cubism in this journal and
Pilate Goes to Kindergarten.
Related material: Chesterton + Thursday in this journal.
Saturday, June 15, 2013
At the Still Point
(Continued from yesterday's posts, "Object of Beauty"
and "Amy's Shadow")
A winner of a Nobel Prize for X-ray crystallography stands
at the head of the New York Times obituary list today.
In memoriam —
X-Ray Vision "Crystal Engineering in Kindergarten," by Bart Kahr:
"If the reader is beginning to suspect that Froebel’s Click images for some backstories. |
Some further background:
The Times follows yesterday's egregious religious error
with an egregious scientific error:
"The technique developed by Dr. Karle and Herbert A. Hauptman,
called X-ray crystallography, is now routinely used by scientists…."
Karle was reportedly born in 1918, Hauptman in 1917.
Wikipedia on the history of X-ray crystallography:
"The idea that crystals could be used as a
diffraction grating for X-rays arose in 1912…."
The Nobel Foundation:
"The Nobel Prize in Physics 1914 was awarded to
Max von Laue 'for his discovery of the diffraction of
X-rays by crystals.'"
"The Nobel Prize in Physics 1915 was awarded jointly to
Sir William Henry Bragg and William Lawrence Bragg
'for their services in the analysis of crystal structure
by means of X-rays.'"
Friday, April 19, 2013
The Large Desargues Configuration
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6-volume
Principles of Geometry .
Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
Monday, April 1, 2013
Desargues via Rosenhain
Background: Rosenhain and Göpel Tetrads in PG(3,2)
Introduction: The Large Desargues Configuration Added by Steven H. Cullinane on Friday, April 19, 2013 Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point The converse, from the same book:
"If two triangles are perspective from a line
Desargues' theorem according to Wikipedia
"Two triangles are in perspective axially [i.e., from a line]
A figure often used to illustrate the theorem,
A discussion of the "if and only if" version of the theorem
This large Desargues configuration involves a third triangle,
Point-line incidence in this larger configuration is,
The third triangle, within the larger configuration,
|
A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desargues-theorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original Veblen-Young figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.
In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):
Volume I of Baker's Principles has a cover closely related to the Rosenhain tetrads in PG(3,2) |
Volume IV of Baker's Principles has a cover closely related to the Göpel tetrads in PG(3,2) |
Foundations (click to enlarge)
|
Higher Geometry (click to enlarge)
|
Wednesday, March 27, 2013
The Barnes School
Randy Kennedy in tomorrow's print edition
of The New York Times—
Art collector Albert C. Barnes "viewed his foundation
less as a museum than as a school."
Roberta Smith in the New York Times
print edition of May 18, 2012, on
art arrangements by Albert C. Barnes—
"Barnes’s arrangements are as eye-opening,
intoxicating and, at times, maddening as ever, maybe more so.
They mix major and minor in relentlessly symmetrical patchworks
that argue at once for the idea of artistic genius and the
pervasiveness of talent. Nearly every room is an exhibition
unto itself— a kind of art wunderkammer, or cabinet of curiosities…."
This journal at noon on the same day, May 18, 2012—
Balakrishnan's Banners
See also Brightness at Noon from March 25.
Monday, March 4, 2013
Occupy Galois Space
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
Friday, March 1, 2013
Spinning in Infinity
(Continued from Jan. 13 and Feb. 19.)
The founder of Graylock Press
died at 96 in Bethesda on Feb. 19:
For some background on the original Bethesda,
see a webpage on Angels in America.
For some background on noted Graylock authors,
see Pavel Alexandrov.
For deeper background, see a book praised by Freeman Dyson:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Friday, February 8, 2013
Dictum
A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields
the following.
Wightman at Wikipedia:
"His graduate students include
Arthur Jaffe, Jerrold Marsden, and Alan Sokal."
"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."
— Arthur Jaffe in News at Princeton , Jan. 30
Marsden at Wikipedia:
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)
A Wikipedia reference in the symplectic geometry article leads to…
THE SYMPLECTIZATION OF SCIENCE:
Mark J. Gotay
James A. Isenberg February 18, 1992 Acknowledgments:
We would like to thank Jerry Marsden and Alan Weinstein Published in: Gazette des Mathématiciens 54, 59-79 (1992). Opening:
"Physics is geometry . This dictum is one of the guiding |
A different account of the dictum:
The strange term Geometrodynamics
is apparently due to Wheeler.
Physics may or may not be geometry, but
geometry is definitely not physics.
For some pure geometry that has no apparent
connection to physics, see this journal
on the date of Wightman's death.
Thursday, October 25, 2012
Bunker Bingo*
Mathematics and Narrative continues…
Steiner's version of "classical functional analysis"—
"Mein Führer… Steiner…"
* See the story by Kilgore Trout. See also On Linguistic Creation,
The Matrix of Abraham, and The Thoreau Foundation.
Tuesday, October 23, 2012
Labyrinth 23
The title refers to a search (see below)
suggested by three things—
- David Foster Wallace biographer D. T. Max—
"There's a note in one of my files where he says something like,
'Infinite Jest was just a means to Mary Karr's end, as it were.' " - "There is a body on the cross in my church ." —Mary Karr
- A body.
The search— Labyrinth 23.
(Within the search results, note particularly the post "The Infinity Point.")
Sunday, October 14, 2012
Crossroads
"Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself."
— A translated remark by Hermann Weyl, p. 136, "The Current Epistemogical Situation in Mathematics" in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, 1998, pp. 123-142, as cited by David Corfield
Corfield once wrote that he would like to know the original German of Weyl's remark. Here it is:
"Die Mathematik ist nicht das starre und Erstarrung bringende Schema, als das der Laie sie so gerne ansieht; sondern wir stehen mit ihr genau in jenem Schnittpunkt von Gebundenheit und Freiheit, welcher das Wesen des Menschen selbst ist."
— Hermann Weyl, page 533 of "Die heutige Erkenntnislage in der Mathematik" (Symposion 1, 1-32, 1925), reprinted in Gesammelte Abhandlungen, Band II (Springer, 1968), pages 511-542
For some context, see a post of January 23, 2006.
Monday, September 17, 2012
The Count
… I saw a shadow
I rose to a knee, |
Simpson reportedly died on Holy Cross Day.
That day in this journal—
Friday, June 29, 2012
The Uploading (continued)*
It Must Be Abstract
It Must Change
It Must Give Pleasure
— Parts of a poem by Wallace Stevens
“At that instant he saw, in one blaze of light, an image of unutterable conviction, the reason why the artist works and lives and has his being–the reward he seeks–the only reward he really cares about, without which there is nothing. It is to snare the spirits of mankind in nets of magic, to make his life prevail through his creation, to wreak the vision of his life, the rude and painful substance of his own experience, into the congruence of blazing and enchanted images that are themselves the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”
– Thomas Wolfe, Of Time and the River
Of Time and the River and the Frogs —
* This post's title refers to the above uploading date— Jan. 26, 2008.
Friday, February 17, 2012
Pregeometry and Finite Geometry
Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .
From a paper on that topic—
"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."
— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science 2 (1): 137-176
Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.
Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.
Some related material from pure mathematics—
Click image for further details.
Sunday, January 29, 2012
Declarations
Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.
My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).
A 1976 document relevant to the concerns in the Trudeau book—
Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.
They apply to finite geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations
of mathematics.
"The hand of the creative artist, laid upon the major premise,
rocks the foundations of the world." — Dorothy Sayers
Thursday, January 19, 2012
Hegel
Those impressed by George Steiner's remark on Hegel in the previous post may consult…
(The Christian Examiner. Volume LXXX. New Series, Volume I. January, March, May, 1866.
New York: James Miller, Publisher, 522, Broadway. Boston: Walker, Fuller, & Co.
No. CCLIV, Art. IV.– THE SECRET OF HEGEL.
By C. C. Everett, pp. 196-207.
A review of…
The Secret of Hegel, being the Hegelian System in Origin, Principle, Form, and Matter.
By James Hutchinson Sterling. In two volumes.
London: Longman, Green, Longman, Roberts, & Green. 1865. 8vo, 2 vols.)
On Hegel, from the review—
"He starts not from the beginning, but from the heart, of the world.
There never was a time when this pure Being— which, in its
undivided absoluteness, is indistinguishable from nothing;
as pure, unbroken light is indistinguishable from darkness—
was by itself alone; but this absolute Being is yet the foundation
and the groundwork of whatever is."
For more on Hegel's logic, see Marxists.org.
See also Steiner on chess and Lenin in The New Yorker
(September 7, 1968, page 133).
Friday, January 13, 2012
Logos
Click logos for related persons.
Background from this journal—
Collegiality, That Hideous Strength , and The Oxford Murders .
See also…
"The heart of the book is the conveying of a meaningful understanding
of where mathematical results originated…."
Friday, December 16, 2011
Take Your Pick
Two recent quotes in this journal—
"Hoban once ruefully observed that death would be a good career move:
'People will say, "Yes, Hoban, he seems an interesting writer, let’s look at him again."'"
"This poem is concerned with language on a very plain level."
— "Paradoxes and Oxymorons" in Shadow Train
Michael Kinsley in The New York Times on Sunday, May 13, 2007—
Kinsley on the career of Christopher Hitchens—
Interesting! …. Interesting!! …. Interesting!!! …. Interesting!!!!
Where was this train heading?
Kinsley on a book in which Hitchens …
… pronounces the movie “It’s a Wonderful Life” “engaging but abysmal” (a typical Hitchens aside: cleverly paradoxical? witlessly oxymoronic? take your pick)….
Thursday, December 15, 2011
As Is
"This poem is concerned with language on a very plain level."
"You got to ride it like you find it."
— Song lyric
Related entertainment —
Monday, October 31, 2011
Beauty, Truth, Halloween
On Halloween…
"Remember that for Ockham there is nothing in the universe that is
in any way universal except a concept or word: there are no real
natures shared by many things. However, things do resemble one
another, some things more closely than others. So the various
degrees of resemblance give a foundation in reality for our conceptual
structures, such as Porphyry's tree.
Now resemblance (or similitude or likeness) is a relation.
If such relations are realities, then we can say that there are realities
out there that correspond to our conceptual structures."
— R.J. Kilcullen at Macquarie University, course labeled Phil360
"The kernel of a homomorphism is always a congruence.
Indeed, every congruence arises as a kernel."
— Congruence Relation, section on Universal Algebra, in Wikipedia
"Beauty then is a relation."
"An Attempt to Understand the Problem of Universals"
is the title of a talk by Fabian Geier, University of Bamberg—
"The talk was held at Gdańsk University on May 26th 2008."
Related material— Stevie Nicks turns 60.
Wednesday, October 26, 2011
Erlanger and Galois
Peter J. Cameron yesterday on Galois—
"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."
Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.
Group theory is an essential part of modern geometry as well as of modern algebra—
"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."
— Felix Christian Klein, Erlanger Programm , 1872
("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))
Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—
"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."
For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.
* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2
Saturday, October 8, 2011
An Ordinary Evening in Hartford
From Rebecca Goldstein's Talks and Appearances page—
• "36 (Bad) Arguments for the Existence of God,"
Annual Meeting of the Freedom from Religion Foundation,
Marriot, Hartford, CT, Oct 7 [2011], 7 PM
From Wallace Stevens—
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
For those who prefer greater depth on Yom Kippur, yesterday's cinematic link suggests…
"Yo sé de un laberinto griego que es una línea única, recta."
—Borges, "La Muerte y la Brújula " ("Death and the Compass")
See also Alpha and Omega (Sept. 18, 2011) and some context from 1931.
Sunday, September 4, 2011
The Found Numbers
(A sequel to The Lost Word)
NY Lottery, Sunday, Sept. 4, 2011
The number "38" might refer, notably, to
- The old postal code of Harvard University,
- the caliber of one type of handgun…
… and, less notably, to a page number
that appears in most books.
T.S. Eliot's essay "Tradition and the Individual Talent"
was first published in 1919.*
For some background on that year, see the Harvard Class of 1919.
For a notable instance of the page number 38, see
Poetry and Belief in the Work of T.S. Eliot ,
by Kristian Smidt (first published in 1949 by
the Norwegian Academy of Science and Letters).
* "Tradition and the Individual Talent" in The Egoist
at the Modernist Journals Project:
Part I in Vol. 6, No. 4 (Sept. 1919),
Parts II-III in Vol. 6, No. 5 (Dec. 1919)
Saturday, August 20, 2011
Castles in the Air
"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace
"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
"King Solomon's Mines," 1937—
The image above is an illustration from "Romancing the Hyperspace," May 4, 2010.
Happy birthday to the late Salomon Bochner.
Wednesday, July 13, 2011
The Sinatra Code
From The Da Vinci Code, Chapter 56 Sophie stared at Teabing a long moment and then turned to Langdon. “The Holy Grail is a person?” Langdon nodded. “A woman, in fact.” From the blank look on Sophie’s face, Langdon could tell they had already lost her. He recalled having a similar reaction the first time he heard the statement. It was not until he understood the symbology behind the Grail that the feminine connection became clear. Teabing apparently had a similar thought. “Robert, perhaps this is the moment for the symbologist to clarify?” He went to a nearby end table, found a piece of paper, and laid it in front of Langdon. Langdon pulled a pen from his pocket. “Sophie are you familiar with the modern icons for male and female?” He drew the common male symbol ♂ and female symbol ♀. “Of course,” she said. “These,” he said quietly, are not the original symbols for male and female. Many people incorrectly assume the male symbol is derived from a shield and spear, while the female represents a mirror reflecting beauty. In fact, the symbols originated as ancient astronomical symbols for the planet-god Mars and the planet-goddess Venus. The original symbols are far simpler.” Langdon drew another icon on the paper. ∧
“This symbol is the original icon for male ,” he told her. “A rudimentary phallus.” “Quite to the point,” Sophie said. “As it were,” Teabing added. Langdon went on. “This icon is formally known as the blade , and it represents aggression and manhood. In fact, this exact phallus symbol is still used today on modern military uniforms to denote rank.” “Indeed.” Teabing grinned. “The more penises you have, the higher your rank. Boys will be boys.” Langdon winced. “Moving on, the female symbol, as you might imagine, is the exact opposite.” He drew another symbol on the page. “This is called the ∨ Sophie glanced up, looking surprised. Langdon could see she had made the connection. “The chalice,” he said, “resembles a cup or vessel, and more important, it resembles the shape of a woman’s womb. This symbol communicates femininity, womanhood, and fertility.” Langdon looked directly at her now. “Sophie, legend tells us the Holy Grail is a chalice—a cup. But the Grail’s description as a chalice is actually an allegory to protect the true nature of the Holy Grail. That is to say, the legend uses the chalice as a metaphor for something far more important.” “A woman,” Sophie said. “Exactly.” Langdon smiled. “The Grail is literally the ancient symbol for womankind, and the Holy Grail represents the sacred feminine and the goddess, which of course has now been lost, virtually eliminated by the Church. The power of the female and her ability to produce life was once very sacred, but it posed a threat to the rise of the predominantly male Church, and so the sacred feminine was demonized and called unclean. It was man , not God, who created the concept of ‘original sin,’ whereby Eve tasted of the apple and caused the downfall of the human race. Woman, once the sacred giver of life, was now the enemy.” “I should add,” Teabing chimed, “that this concept of woman as life-bringer was the foundation of ancient religion. Childbirth was mystical and powerful. Sadly, Christian philosophy decided to embezzle the female’s creative power by ignoring biological truth and making man the Creator. Genesis tells us that Eve was created from Adam’s rib. Woman became an offshoot of man. And a sinful one at that. Genesis was the beginning of the end for the goddess.” “The Grail,” Langdon said, “is symbolic of the lost goddess. When Christianity came along, the old pagan religions did not die easily. Legends of chivalric quests for the lost Grail were in fact stories of forbidden quests to find the lost sacred feminine. Knights who claimed to be “searching for the chalice” were speaking in codes as a way to protect themselves from a Church that had subjugated women, banished the Goddess, burned nonbelievers, and forbidden pagan reverence for the sacred feminine.” |
Happy birthday to Harrison Ford.
One for my baby…
∧
One more for the road.
∨
Sunday, March 27, 2011
Frames
"I just seemed to have more frames per second than other kids."
— Mary Karr, "Facing Altars: Poetry and Prayer"
See also "Signs and Symbols."
Sunday, March 13, 2011
Twenty-Four
"Poetry never left me stranded, and as an atheist most of my life, I presumed its mojo was a highbrow, intellectual version of what religion did for those more gullible believers in my midst— dumb bunnies to a one, the faithful seemed to me, till I became one.
In the Texas oil town where I grew up, fierceness won fights, but I was thin-skinned— an unfashionably bookish kid whose brain wattage was sapped by a consuming inner life others didn’t seem to bear the burden of. I just seemed to have more frames per second than other kids."
— "Facing Altars: Poetry and Prayer," by Mary Karr
"The original movie had been slowed to a running time of twenty-four hours.
What he was watching seemed pure film, pure time.
The broad horror of the old gothic movie was subsumed in time."
— Point Omega , by Don DeLillo
Thursday, December 2, 2010
Caesarian
The Dreidel Is Cast
The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."
That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.
The holiday is celebrated with, among other things, the Jewish version of a die— the dreidel . Note the similarity of the dreidel to an illustration of The Stone* on the cover of the 2001 Eerdmans edition of Charles Williams's 1931 novel Many Dimensions—
For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.
Here is the opening of Many Dimensions—
For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.
Some further holiday meditations on a beginning—
Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography—
"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-
those branches of mathematics in which I had been personally involved.
He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-
The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."
For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.
The following remarks by Hyland are of more general interest—
"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."
— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.
Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota—
See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.
* Williams's novel says the letters of The Stone are those of the Tetragrammaton— i.e., Yod, He, Vau, He (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone include the three-pronged letter Shin , also found on the dreidel . What esoteric religious meaning is implied by this, I do not know.
Monday, October 4, 2010
The Search for Wisdom
"Even before Thales fell into the well, and the ancient Greeks laid the foundation
for Western philosophy, humans were engaged in the search for wisdom…."
— Introducing "The Stone," NY Times, May 16, 2010, quoted here this morning
Some have Wisdom thrust upon them.
Context— Happy Birthday Reba, 2009, and "I'll Follow a Star."
Thursday, July 22, 2010
Soul Riff
A link in the previous post to the Harvard Foundation led to a screenshot
of a long-neglected Harvard web page. That page stated yesterday that
"the Harvard Foundation 2008 Artist of the Year has not yet been announced."
It turns out to be jazz artist Herbie Hancock, who was honored at
Harvard's Sanders Theatre on Saturday, March 1, 2008.
Related material from this journal—
For a response from the next day,
March 1, click on the professor.
Wednesday, July 21, 2010
Dream Names continued
From the 7/20 Harvard Crimson—
"The scholarly expeditions undertaken by modern-day explorer and Harvard Foundation Director S. Allen Counter will be featured in a biopic produced by actor Will Smith.
…. Debbie Allen is also producing the film, and Farhad Safinia will be penning the script, Variety magazine reported.
…. Counter said that Debbie Allen described his character as 'a mixture of Indiana Jones and Robert Langdon,' the fictional Harvard professor of symbology in Dan Brown’s novels."
Farhad Safinia is co-writer and co-producer, with Mel Gibson, of "Apocalypto."
From "The Envelope: The Awards Insider" at the LA Times, a review of the film based on Dan Brown's "Angels & Demons"—
"The script tips its hand too early, and can't quite turn Langdon into Indiana Langdon on his Last Crusade."
— May 15, 2009
, Orlando Sentinel movie critic,Related material:
The Robert Jones Code—
Monday, June 28, 2010
Shall I Compare Thee
Margaret Soltan on a summer's-day poem by D.A. Powell—
first, a congregated light, the brilliance of a meadowland in bloom
and then the image must fail, as we must fail, as we
graceless creatures that we are, unmake and befoul our beds
don’t tell me deluge. don’t tell me heat, too damned much heat
"Specifically, your trope is the trope of every life:
the organizing of the disparate parts of a personality
into a self (a congregated light), blazing youth
(a meadowland in bloom), and then the failure
of that image, the failure of that self to sustain itself."
Alternate title for Soltan's commentary, suggested by yesterday's Portrait:
Smart Jewish Girl Fwows Up.
Midrash on Soltan—
Wert thou my enemy, O thou my friend,
How wouldst thou worse, I wonder, than thou dost
Defeat, thwart me?
"…meadow-down is not distressed
For a rainbow footing…."
Tuesday, June 22, 2010
Mathematics and Narrative, 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
A 1973 review of Koestler's book—
"Koestler's 'call girls,' summoned here and there
by this university and that foundation
to perform their expert tricks, are the butts
of some chilling satire."
Examples of Light—
Felix Christian Klein (1849- June 22, 1925) and Évariste Galois (1811-1832)
Klein on Galois—
"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."
— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.
"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."
— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.
Examples of Darkness—
Martin Gardner on Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
Gardner was reviewing a recent book about Galois by one Amir Alexander.
Alexander himself has written some reviews relevant to the Koestler book above.
See Alexander on—
The 2005 Mykonos conference on Mathematics and Narrative
A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."
Sunday, June 20, 2010
Lovasz Wins Kyoto Prize
From a June 18 press release—
KYOTO, Japan, Jun 18, 2010 (BUSINESS WIRE) — The non-profit Inamori Foundation (President: Dr. Kazuo Inamori) today announced that Dr. Laszlo Lovasz will receive its 26th annual Kyoto Prize in Basic Sciences, which for 2010 focuses on the field of Mathematical Sciences. Dr. Lovasz, 62, a citizen of both Hungary and the United States, will receive the award for his outstanding contributions to the advancement of both the academic and technological possibilities of the mathematical sciences.
Dr. Lovasz currently serves as both director of the Mathematical Institute at Eotvos Lorand University in Budapest and as president of the International Mathematics Union. Among many positions held throughout his distinguished career, Dr. Lovasz also served as a senior research member at Microsoft Research Center and as a professor of computer science at Yale University.
Related material: Cube Space, 1984-2003.
See also “Kyoto Prize” in this journal—
The Kyoto Prize is “administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”
— – Montreal Gazette, June 20, 2008
Friday, June 11, 2010
The Raven
"Why is a raven like a writing-desk?" — Alice's Adventures in Wonderland
Arthur Koestler, The Roots of Coincidence —
"In his The Nature of the Physical World (1928) Sir Arthur Eddington introduced his famous 'parable of the two writing desks.' One is the antique piece of furniture on which his elbows solidly rest while writing; the other is the desk as the physicist conceives it, consisting almost entirely of empty space, sheer nothingness…. Eddington concluded:
In the world of physics we watch a shadowgraph performance of familiar life. The shadow of my elbow rests on the shadow-table as the shadow-ink flows over the shadow-paper….
Though the constituents of matter could be described with great mathematical accuracy as patterns of vibrations, the question remained— what was it that vibrated? On the one hand, these matter-waves produced physically real phenomena, such as interference patterns on a screen, or the currents in a transistor radio. On the other hand, the whole conception of matter-waves excludes by definition any medium with physical attributes as a carrier of the waves. A wave is movement; but what is that something that moves, producing the shadows on Eddington's shadow-desk? Short of calling it the grin of the Cheshire Cat, it was named the 'psi field' or 'psi function.'"
What is it that moves? Perhaps not the Cheshire Cat, but rather The Raven—
Closeup, he’s blue—streaked iris blue, india-ink blue—and
black—an oily, fiery set of blacks—none of them
true—as where hate and order touch—something that cannot
become known. Stages of black but without
graduation. So there is no direction.
All of this happened, yes.
— Jorie Graham, "The Dream of the Unified Field"
See also notes on darkness in this journal.
Thursday, June 10, 2010
The Celestine Dream
"…after all, the word 'bolshoi' means big, great or grand."
"How much for a dream?" — This journal, January 1, 2010
Star of Bolshoi Ballet Dies at 102
I was eight—
I saw the different weights of things,
saw the vivid performance of the present,
saw the light rippling almost shuddering where her body finally
touched
the image, the silver film between them like something that would have
shed itself in nature now
but wouldn’t, couldn’t, here, on tight,
between, not thinning, not slipping off to let some
seed-down
through, no signal in it, no information …
— Jorie Graham, "The Dream of the Unified Field"
Update of 10:25 AM June 10— Click to enlarge—
Friday, January 1, 2010
Thursday, November 19, 2009
Walden for Jews
“Orthodox Jews are disappearing from Jerusalem. One moment they are praying at the Western Wall, and in the blink of an eye, they seem to evaporate…. In order to build the Third Temple while being respectful of the Islamic structures on the Temple Mount, the Jews have discovered a way to access a fourth spatial dimension. They will build the Third Temple invisibly ‘above’ the Temple Mount and ‘above’ the Mosque in the direction of the fourth dimension.”
— Clifford Pickover, description of his novel Jews in Hyperspace
“If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.”
— Henry David Thoreau, conclusion of Walden
Related material: Log24 entries, morning and evening of June 11, 2009, “Text” (June 22, 2009), and Salomon Bochner‘s remarks on space in “Eight is a Gate” (Feb. 26, 2008).
Monday, February 23, 2009
Monday February 23, 2009
McGee and Smee
Project MUSE — … and interpretations, “any of the Zingari shoolerim [gypsy schoolchildren] may pick a peck of kindlings yet from the sack of auld hensyne” (FW 112.4-8). … — Patrick McGee, “Reading Authority: |
“The ulterior motive behind this essay [“Reading Authority,” above], the purpose for which I seize this occasion, concerns the question or problem of authority. I stress at the outset my understanding of authority as the constructed repository of value or foundation of a system of values, the final effect of fetishism– in this case, literary fetishism. [Cf. Marx, Das Kapital] Reading– as in the phrase ‘reading authority’– should be grasped as the institutionally determined act of constructing authority….”
“[In Peter Pan] Smee is Captain Hook’s right-hand man… Barrie describes him as ‘Irish’ and ‘a man who stabbed without offence‘….”
Background: In yesterday’s morning entry, James Joyce as Jesuit, with “Dagger Definitions.”
A different Smee appears as an art critic in yesterday’s afternoon entry “Design Theory.”–
“Brock, who has a brisk mind, is a man on a mission. He read mathematical economics and political philosophy at Princeton (he has five degrees in all) and is the founder and president of Strategic Economic Decisions Inc., a think tank specializing in applying the economics of uncertainty to forecasting and risk assessment.
But phooey to all that; Brock has deeper things to think about. He believes he has cracked the secret of beautiful design. He even has equations and graphs to prove it.”
A Jesuit in Portrait of the Artist as a Young Man:
“When may we expect to have something from you on the esthetic question?”
“Our entanglement in the wilderness of Finnegans Wake is exemplified by the neologism ‘Bethicket.’ This word condenses a range of possible meanings and reinforces a diversity of possible syntactic interpretations. Joyce seems to allude to Beckett, creating a portmanteau word that melds ‘Beckett’ with ‘thicket’ (continuing the undergrowth metaphor), ‘thick’ (adding mental density to floral density)…. As a single word ‘Bethicket’ contains the confusion that its context suggests. On the one hand, ‘Bethicket me for a stump of a beech’ has the sound of a proverbial expletive that might mean something like ‘I’ll be damned’ or ‘Well, I’ll be a son of a gun.’….”
At the Oscars, 2009
Related material:Frame Tales and Dickung
Sunday, December 7, 2008
Sunday December 7, 2008
the Soul
On a book by Margaret Wertheim:
“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)
There are also other sorts of space.
© 2005 The Institute for Figuring
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
This photo may serve as an
introduction to a different
sort of space.
See The Eightfold Cube.
For the religious meaning
of this small space, see
the eight I Ching trigrams.
For a related larger space,
see the entry and links of
St. Augustine’s Day, 2006.
Wednesday, November 26, 2008
Wednesday November 26, 2008
Recent abstracts of interest:
Kuwait Foundation Lectures —
Jan. 29, 2008: J. P. Wintenberger, “On the Proof of Serre’s Conjecture“
Oct. 28, 2008: Chandrashekhar Khare, “Modular Forms and Galois Representations“
Background:
The Last Theorem, a novel by Arthur C. Clarke and Frederik Pohl published Aug. 5, 2008
“Going Beyond Fermat’s Last Theorem,” a news article in The Hindu published April 25, 2005
Wikipedia: Serre Conjecture (Number Theory)
Henri Darmon, “Serre’s Conjectures“
Tuesday, August 5, 2008
Tuesday August 5, 2008
The Last Theorem, a novel by
Arthur C. Clarke and Frederik Pohl
"The Last Theorem is a story of one man’s mathematical obsession, and a celebration of the human spirit and the scientific method. It is also a gripping intellectual thriller….
In 1637, the French mathematician Pierre de Fermat scrawled a note in the margin of a book about an enigmatic theorem: 'I have discovered a truly marvelous proof of this proposition which this margin is too narrow to contain.' He also neglected to record his proof elsewhere. Thus began a search for the Holy Grail of mathematics– a search that didn’t end until 1994, when Andrew Wiles published a 150-page proof. But the proof was burdensome, overlong, and utilized mathematical techniques undreamed of in Fermat’s time, and so it left many critics unsatisfied– including young Ranjit Subramanian, a Sri Lankan with a special gift for mathematics and a passion for the famous 'Last Theorem.'
When Ranjit writes a three-page proof of the theorem that relies exclusively on knowledge available to Fermat, his achievement is hailed as a work of genius, bringing him fame and fortune…."
For a similar third-world fantasy about another famous theorem, see the oeuvre of Ashay Dharwadker.
Note the amazing conclusion of Dharwadker's saga (thus far)–
Dharwadker devises a proof of the four-color theorem that leads to…
of the Standard Model
with Quantum Gravity!
and Usenet Postings.
For another connection with Sri Lanka, see
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, January 3, 2008
Thursday January 3, 2008
continued:
The Revelation Game
New Year’s reading for
the tigers of Princeton
Two reviews from the February 2008 Notices of the American Mathematical Society:
From a review of
A Certain Ambiguity
(A Mathematical Novel)
by Gaurav Suri and Hartosh Singh Bal
Princeton University Press
Hardcover, US$27.95, 281 pages —
“From the Habermas-Lyotard debate (see [1] for an introduction) to the Sokal hoax ([4]), to recent atheist manifestos on the bestseller lists (e.g., [2]) the question of foundations for intellectual thought and especially for intellectual debate has never been more critical or urgent.”
[1] M. Bérubé, What’s Liberal about the Liberal Arts? Classroom Politics and “Bias” in Higher Education, W. W. Norton, 2006.
[2] S. Harris, Letter to a Christian Nation, Knopf, 2006.
[4] A. Sokal and P. Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, 1999.
Also in the February Notices– a review of a book, Superior Beings: If They Exist, How Would We Know?, in which the author
“.. uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….
In the ‘Revelation Game,’ for example,
the Person (P) has two options:
1) P can believe in SB’s existence
2) P can not believe in SB’s existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself….
… [and] goals allow us to rank all the outcomes for each player from best… to worst…. The question we must answer is: what is the Nash equilibrium in this case?”
The answer is what one might expect from the American Mathematical Society:
“… the dominant strategy for both is when SB does not reveal Himself and P does not believe in His existence.”
Other strategies are, of course, possible. See last year’s entries.
See also
the life of John Nash,
for whom the above
equilibrium is named.
Thursday, November 29, 2007
Thursday November 29, 2007
From today's online NY Times:
Obituaries in the News
By THE ASSOCIATED PRESS
Published: [Wednesday] Gennie DeWeese BOZEMAN, Mont. (AP) — Gennie DeWeese, an artist known for her landscape paintings and woodblock prints whose works are displayed at museums across the Northwest, died Monday [November 26, 2007]. She was 86. DeWeese died at her studio south of Bozeman. Dahl Funeral Chapel confirmed her death. Her first oil painting was of her dog, done when she was 12 years old. In 1995, DeWeese received an honorary doctorate of fine arts from Montana State University, and she received the Montana Governor's Award for the Arts. |
Robert M. Pirsig in
Zen and the Art of Motorcycle Maintenance
(April 1974) —
"The rhetoricians of ancient Greece were the first teachers in the history of the Western world. Plato vilified them in all his works to grind an axe of his own and since what we know about them is almost entirely from Plato they’re unique in that they’ve stood condemned throughout history without ever having their side of the story told. The Church of Reason that I talked about was founded on their graves. It’s supported today by their graves. And when you dig deep into its foundations you come across ghosts." I look at my watch. It’s after two. "It’s a long story," I say. "You should write all this down," Gennie says. |
Quod erat
demonstrandum.
For more information,
click on the black monolith.
Related material:
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
Tuesday, October 16, 2007
Tuesday October 16, 2007
Harish-Chandra,
who died at 60
on this date in 1983
Harish-Chandra in 1981
(Photo by Herman Landshof)
As a corrective to the previous parodies here, the following material on the mathematician Harish-Chandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.
MacTutor History of Mathematics:
“Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.”
R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):
“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semi-simple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”
“It is difficult to communicate the grandeur of Harish-Chandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”
For deeper views of his work, see
- Rebecca A. Herb, “Harish-Chandra and His Work” (pdf), Bulletin of the American Mathematical Society, July 1991, and
- R. P. Langlands, “Harish-Chandra, 1923-1983” (pdf, 28 pp., Royal Society memoir, 1985)
Thursday, October 11, 2007
Thursday October 11, 2007
Comments today on Peter Woit’s weblog entry “Deep Beauty“–
Sunday, October 7, 2007
Sunday October 7, 2007
was the title of a symposium on quantum theory at Princeton last week dedicated to the late John von Neumann. The title was left undefined. In honor of von Neumann, here is some material that may help those searching for the title’s meaning:
at Arxiv Structure
of a paper titled
“Quantum Theory From
Five Reasonable Axioms.”
Those who wisely object that popularity should not be a test of beauty may consult a little-known (at least in the West) Sino-Japanese definition of “deep beauty.” This definition– although from philosophy, not physics– may appeal to those who, like Peter Woit, are troubled by a Christian foundation’s sponsorship of last week’s scientific symposium.
“Deep beauty”
is yuugen.
Thursday, September 27, 2007
Thursday September 27, 2007
continues:
Classics 101 —
(See September 15. )
"The communication
of the dead is tongued with fire
beyond the language of the living."
— T. S. Eliot, Four Quartets
The Boston Globe,
Tuesday, Sept. 25, 2007-
Saturday, August 18, 2007
Saturday August 18, 2007
“What on earth is
a ‘concrete universal’?“
— Said to be an annotation
(undated) by Robert M. Pirsig
of A History of Philosophy,
by Frederick Copleston,
Society of Jesus.
“No matter how it’s done,
you won’t like it.“
— Robert Redford to
Robert M. Pirsig in Lila
“In chapters 19 and 20 of LILA there is a discussion about the possibility of making Zen and the Art into a movie. It opens with a scene where Robert Redford, who ‘really would like to have the film rights,’ comes to meet and negotiate with Phaedrus in his New York City hotel room. Phaedrus tells the famous actor that he can have the rights to the book, but maybe that’s just because he’s star-struck and doesn’t like to haggle. Under his excitement, Phaedrus has a bad feeling about it. He tells us that he’s been warned by several different people not to allow such a film to be made. Even Redford warned him not to do it. So what’s the problem? As it’s put at the end of that discussion, ‘Films are social media; his book was largely intellectual. That was the center of the problem.'”
— David Buchanan at robertpirsig.org
“The insight is constituted precisely by ‘seeing’ the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete.”
— Fr. Brian Cronin‘s Foundations of Philosophy, Ch. 2, “Identifying Direct Insights,” quoted in Ideas and Art
See also Smiles of a Summer Evening, the current issue of TIME, the time of this entry (7:20:11 PM ET), and Plato, Pegasus, and the Evening Star.
Saturday, July 14, 2007
Saturday July 14, 2007
Catholic University
of America
The August 2007 issue of Notices of the American Mathematical Society contains tributes to the admirable personal qualities and mathematical work of the late Harvard professor George Mackey. For my own tributes, see Log24 on March 17, 2006, April 29, 2006, and March 10, 2007. For an entry critical of Mackey’s reductionism– a philosophical, not mathematical, error– see Log24 on May 23, 2007 (“Devil in the Details”).
Here is another attack on reductionism, from a discussion of the work of another first-rate mathematician, the late Gian-Carlo Rota of MIT:
“Another theme developed by Rota is that of ‘Fundierung.’ He shows that throughout our experience we encounter things that exist only as founded upon other things: a checkmate is founded upon moving certain pieces of chess, which in turn are founded upon certain pieces of wood or plastic. An insult is founded upon certain words being spoken, an act of generosity is founded upon something’s being handed over. In perception, for example, the evidence that occurs to us goes beyond the physical impact on our sensory organs even though it is founded upon it; what we see is far more than meets the eye. Rota gives striking examples to bring out this relationship of founding, which he takes as a logical relationship, containing all the force of logical necessity. His point is strongly antireductionist. Reductionism is the inclination to see as ‘real’ only the foundation, the substrate of things (the piece of wood in chess, the physical exchange in a social phenomenon, and especially the brain as founding the mind) and to deny the true existence of that which is founded. Rota’s arguments against reductionism, along with his colorful examples, are a marvelous philosophical therapy for the debilitating illness of reductionism that so pervades our culture and our educational systems, leading us to deny things we all know to be true, such as the reality of choice, of intelligence, of emotive insight, and spiritual understanding. He shows that ontological reductionism and the prejudice for axiomatic systems are both escapes from reality, attempts to substitute something automatic, manageable, and packaged, something coercive, in place of the human situation, which we all acknowledge by the way we live, even as we deny it in our theories.”
— Robert Sokolowski, foreword to Rota’s Indiscrete Thoughts
Fr. Robert Sokolowski, Ph.D., is Professor of Philosophy at The Catholic University of America in Washington, D.C. Ordained a Roman Catholic priest in 1962, he is internationally recognized and honored for his work in philosophy, particularly phenomenology. In 1994, Catholic University sponsored a conference on his work and published several papers and other essays under the title, The Truthful and the Good, Essays In Honor of Robert Sokolowski. |
The tributes to Mackey are contained in the first of two feature articles in the August 2007 AMS Notices. The second feature article is a review of a new book by Douglas Hofstadter. For some remarks related to that article, see Thursday’s Log24 entry “Not Mathematics but Theology.”
Thursday, July 5, 2007
Thursday July 5, 2007
“Although it may not at first be obvious,
the substitution for real religions
of a religion drained of particulars
is of a piece with the desire to
exorcise postmodernism.”
The previous entry linked to an entry of June 2002 that attacked the nominalism of Stanley Fish. Here is another such attack:
From “Stanley Fish: The Critic as Sophist,” by R.V. Young, in Modern Age, June 22, 2003:
In one of the definitive works of conservatism in the twentieth century, Richard Weaver designates the rise of nominalism as a critical turn in the emergence of the intellectual and cultural disintegration associated with liberalism, which it is the business of a reviving conservatism to contest: “The defeat of logical realism in the great medieval debate was the crucial event in the history of Western culture; from this flowed those acts which issue now in modern decadence.” It is nominalism that provides the intellectual foundation– if a paradox may be hazarded– for the attack by Fish and numerous others (their name is Legion) on the very idea of intellectual foundations:
(4). Ideas Have Consequences (Chicago and London, 1948), 3. R.V. YOUNG is Professor of English at North Carolina State University and author of At War With the Word and Doctrine and Devotion in Seventeenth-Century Poetry (2000). |
Simon Blackburn on
Plato and sophists,
realism and nominalism
(previous entry)
Sunday, February 18, 2007
Sunday February 18, 2007
in Harvard Iconology
The next novel starring
Robert Langdon, Harvard author
of "the renowned collegiate
texbook Religious Iconology"
is said to be titled
The Solomon Key.
Related material–
The Harvard Crimson online:
Fishburne To Receive Honors at Cultural Rhythms Acclaimed actor and humanitarian chosen as the Harvard Foundation's Artist of the Year By DORIS A. HERNANDEZ Friday, February 16, 2007 9:37 PM Tony and Emmy Award-winning actor Laurence Fishburne will take the stage later this month as the 2007 Artist of the Year during the 22nd annual Cultural Rhythms festival, the Harvard Foundation for Intercultural and Race Relations announced Friday afternoon. |
Fishburne
as Morpheus
"Metaphor for Morphean morphosis,
Dreams that wake, transform, and die,
Calm and lucid this psychosis,
Joyce's nightmare in Escher's eye….
Dabo claves regni caelorum. By silent shore
Ripples spread from castle rock. The metaphor
For metamorphosis no keys unlock."
— Steven H. Cullinane,
November 7, 1986,
"Endgame"
More on metamorphosis–
Cat's Yarn
(Log24, June 20, 2006):
"The end is where
we start from."
plus.maths.org
and
Garfield 2003-06-24
See also:
Zen Koan
and
Blue Dream.
Update of 5:24 PM
Feb. 18, 2007:
A Xanga footprint from France
this afternoon (3:47 PM EST)
indicates that someone there
may be interested in the above
poem's "claves regni caelorum."
The visitor from France viewed
"Windmills" (Nov. 15, 2005).
Material related to that entry
may be found in various places
at Log24.com. See particularly
"Shine On, Hermann Weyl," and
entries for Women's History
Month last year that include
"Christ at the Lapin Agile."
Monday, January 29, 2007
Monday January 29, 2007
By Indirections
(Hamlet, II, i)
“Michael Taylor (1971)…. contends that the central conflict in Hamlet is between ‘man as victim of fate and as controller of his own destiny.'”– The Gale Group, Shakespearean Criticism, Vol. 71, at eNotes
Doonesbury today:
“Personality is a synthesis of possibility and necessity.”– Soren Kierkegaard
On Fate (Necessity),
Freedom (Possibility),
and Machine Personality–
Part I: Google as Skynet
George Dyson–
The Godel-to-Google Net [March 8, 2005]
A Cathedral for Turing [October 24, 2005]
Dyson: “The correspondence between Google and biology is not an analogy, it’s a fact of life.”
Part II: The Galois Connection
David Ellerman–
“A Theory of Adjoint Functors– with some Thoughts about their Philosophical Significance” (pdf) [November 15, 2005]
Ellerman: “Such a mechanism seems key to understanding how an organism can perceive and learn from its environment without being under the direct stimulus control of the environment– thus resolving the ancient conundrum of receiving an external determination while exercising self-determination.”
For a less technical version, see Ellerman’s “Adjoints and Emergence: Applications of a New Theory of Adjoint Functors” (pdf).
Ellerman was apparently a friend of, and a co-author with, Gian-Carlo Rota. His “theory of adjoint functors” is related to the standard mathematical concepts known as profunctors, distributors, and bimodules. The applications of his theory, however, seem to be less to mathematics itself than to a kind of philosophical poetry that seems rather closely related to the above metaphors of George Dyson. For a less poetic approach to related purely mathematical concepts, see, for instance, the survey Practical Foundations of Mathematics by Paul Taylor (Cambridge University Press, 1999). For less poetically appealing, but perhaps more perspicuous, extramathematical applications of category theory, see the work of, for instance, Joseph Goguen: Algebraic Semiotics and Information Integration, Databases, and Ontologies.
Tuesday, November 7, 2006
Tuesday November 7, 2006
rev. ed., MIT Press, 1978–
said to have been written
on September 23, 1937
For clues to interpreting
today’s Keystone State
mid-day lottery number,
023, see
The Prime Cut Gospel.
For clues to interpreting
today’s Keystone State
evening lottery number,
666, see
the “Apocalypse Now”
quotations on
All Saints’ Day, 2006.
Sunday, July 16, 2006
Sunday July 16, 2006
Mathematics and Narrative
continued…
“Now, at the urging of the UC Berkeley cognitive linguist George Lakoff, liberal America’s guru of the moment, progressive Democrats are practicing to get their own reluctant mouths around some magical new vocabulary, in the hope of surviving and eventually overcoming the age of Bush.”
— Marc Cooper in The Atlantic Monthly, April 2005, “Thinking of Jackasses: The Grand Delusions of the Democratic Party”
Cooper’s “now” is apparently still valid. In today’s New York Times, the leftist Stanley Fish reviews Talking Right, by leftist Geoffrey Nunberg:
“… the right’s language is now the default language for everyone.
On the way to proposing a counterstrategy (it never really arrives), Nunberg pauses to engage in a polite disagreement with his fellow linguist George Lakoff, who has provided a rival account of the conservative ascendancy. Lakoff argues that Republicans have articulated– first for themselves and then for others– a conceptual framework that allows them to unite apparently disparate issues in a single coherent worldview … woven together not in a philosophically consistent framework but in a narrative ‘that creates an illusion of coherence.’
Once again, the Republicans have such a narrative– ‘declining patriotism and moral standards, the out-of-touch media and the self-righteous liberal elite … minorities demanding special privileges … disrespect for religious faith, a swollen government’– but ‘Democrats and liberals have not offered compelling narratives that could compete’ with it. Eighty pages later he is still saying the same thing. ‘The Democrats need a compelling narrative of their own.'”
Lakoff is the co-author of a book on the philosophy of mathematics, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. From Wikipedia’s article on Lakoff:
“According to Lakoff, even mathematics itself is subjective to the human species and its cultures: thus ‘any question of math’s being inherent in physical reality is moot, since there is no way to know whether or not it is.’ Lakoff and Rafael E. Nunez (2000) argue at length that mathematical and philosophical ideas are best understood in light of the embodied mind. The philosophy of mathematics ought therefore to look to the current scientific understanding of the human body as a foundation ontology, and abandon self-referential attempts to ground the operational components of mathematics in anything other than ‘meat.'”
For a long list of related leftist philosophy, see The Thinking Meat Project.
Democrats seeking narratives may also consult The Carlin Code and The Prime Cut Gospel.
Sunday, June 4, 2006
Sunday June 4, 2006
Jeremy Pearce in this morning's New York Times:
"Dr. Fritz Klein, a psychiatrist and sex researcher who studied bisexuals and their relationships and later helped start a foundation for promoting bisexual culture, died on May 24 at his home in San Diego. He was 73. The cause was a heart attack, said his companion, Tom Reise."
(Click to see the larger original,
a photo by Michael Trezzi)
"The Waste Land," |
|
The sea was calm, your heart would have responded |
420 |
Gaily, when invited, beating obedient | |
To controlling hands | |
I sat upon the shore | |
Fishing, with the arid plain behind me | |
Shall I at least set my lands in order? | 425 |
Eliot's note on line 424: |
|
"The Fisher King,"
"Did you lose your mind
"Well, I'm a singer by trade.
Then one night, in the What does all this mean?
I mean, that, |
edited by Phil Stubbs
Saturday, March 25, 2006
Saturday March 25, 2006
In memory of Rolf Myller,
who died on Thursday,
March 23, 2006, at
Mount Sinai Hospital
in Manhattan:
Myller was,
according to the
New York Times,
an architect
whose eclectic pursuits
included writing
children’s books,
The Bible Puzzle Book, and
Fantasex: A Book of Erotic Games.
He also wrote, the Times says,
“Symbols and Their Meaning
(1978), a graphic overview of
children’s nonverbal communication.”
This is of interest in view of the
Log24 reference to “symbol-mongers”
on the date of Myller’s death.
In honor of Women’s History Month
and of Myller’s interests in the erotic
and in architecture, we present
the following work from a British gallery.
This work might aptly be
retitled “Brick Shithouse.”
Related material:
(1) the artist’s self-portrait
and, in view of the cover
illustration for Myller’s
The Bible Puzzle Book,
(2) the monumental treatise
by Leonard Shlain
The Alphabet Versus
the Goddess: The Conflict
Between Word and Image.
Liza Lou 03.03.06 – 08.04.06 White Cube is pleased to present the first UK solo exhibition by Los Angeles-based artist Liza Lou. Combining visionary, conceptual and craft approaches, Lou makes mixed-media sculptures and room-size installations that are suggestive of a transcendental reality. Lou’s work often employs familiar, domestic forms, crafted from a variety of materials such as steel, wood, papier-mâché and fibreglass, which is then covered with tiny glass beads that are painstakingly applied, one at a time, with tweezers. Dazzling and opulent and constantly glistening with refracted light, her sculptures bristle with what Peter Schjeldahl has aptly described as ‘surreal excrescence’. This exhibition, a meditation on the vulnerability of the human body and the architecture of confinement, will include several new figurative sculptures as well as two major sculptural installations. Security Fence (2005) is a large scale cage made up of four steel, chain link walls, topped by rings of barbed wire and Cell (2004-2006), as its name suggests, is a room based on the approximate dimensions of a death row prison cell, a kind of externalized map of the prisoner’s mind. Both Security Fence and Cell, like Lou’s immense earlier installations Kitchen (1991-1995) and Back Yard (1995-1999) are characterized by the absence of their real human subject. But whereas the absent subject in Kitchen and Back Yard could be imagined through the details and accessories carefully laid out to view, in Lou’s two new installations the human body is implied simply through the empty volume created by the surrounding architecture. Both Cell and Security Fence are monochromatic and employ iconic forms that make direct reference to Minimalist art in its use of repetition, formal perfection and materiality. In contrast to this, the organic form of a gnarled tree trunk, Scaffold (2005-2006), its surface covered with shimmering golden beads, juts directly out from the wall. Lou’s work has an immediate ‘shock’ content that works on different levels: first, an acknowledgement of the work’s sheer aesthetic impact and secondly the slower comprehension of the labour that underlies its construction. But whereas in Lou’s earlier works the startling clarity of the image is often a counterpoint to the lengthy process of its realization, for the execution of Cell, Lou further slowed down the process by using beads of the smallest variety with their holes all facing up in an exacting hour-by-hour approach in order to ‘use time as an art material’. Concluding this body of work are three male figures in states of anguish. In The Seer (2005-2006), a man becomes the means of turning his body back in on himself. Bent over double, his body becomes an instrument of impending self-mutilation, the surface of his body covered with silver-lined beads, placed with the exactitude and precision of a surgeon. In Homeostasis (2005-2006) a naked man stands prostrate with his hands up against the wall in an act of surrender. In this work, the dissolution between inside and outside is explored as the ornate surface of Lou’s cell-like material ‘covers’ the form while exposing the systems of the body, both corporeal and esoteric. In The Vessel (2005-2006), Christ, the universal symbol of torture and agony holds up a broken log over his shoulders. This figure is beheaded, and bejewelled, with its neck carved out, becoming a vessel into which the world deposits its pain and suffering. Lou has had numerous solo exhibitions internationally, including Museum Kunst Palast, Düsseldorf, Henie Onstad Kunstsenter, Oslo and Fondació Joan Miró, Barcelona. She was a 2002 recipient of the MacArthur Foundation Fellowship. Liza Lou’s film Born Again (2004), in which the artist tells the compelling and traumatic story* of her Pentecostal upbringing in Minnesota, will be screened at 52 Hoxton Square from 3 – 25 March courtesy of Penny Govett and Mick Kerr. Liza Lou will be discussing her work following a screening of her film at the ICA, The Mall, London on Friday 3 March at 7pm. Tickets are available from the ICA box office (+ 44 (0) 20 7930 3647). A fully illustrated catalogue, with a text by Jeanette Winterson and an interview with Tim Marlow, will accompany the exhibition. White Cube is open Tuesday to Saturday, 10.00 am to 6.00 pm.
For further information please contact Honey Luard or Susannah Hyman on + 44 (0) 20 7930 5373 * Warning note from Adrian Searle |
Monday, February 13, 2006
Monday February 13, 2006
As yesterday’s Lincoln’s Birthday entry indicated, my own sympathies are not with the “created equal” crowd. Still, the Catholic Fascism of Franco admirer Andrew Cusack seems somewhat over-the-top. A more thoughtful approach to these matters may be found in a recommendation by Ross Douthat at The American Scene:
Read Eve Tushnet on the virtues of The Man in the High Castle.
Related material: Log24 on Nov. 14, Nov. 15, and Nov. 16, 2003.
Another item of interest from Eve:
“Transubstantiation [is equivalent but not equal to] art (deceptive accident hides truthful substance), as vs. Plato’s condemnation of the physical & the fictive? (Geo. Steiner)”
Related material:
(excerpt)
by Father Richard John Neuhaus,
First Things 115 (Aug.-Sept. 2001), 47-56:
“In Grammars of Creation, more than in his 1989 book Real Presences, Steiner acknowledges that his argument rests on inescapably Christian foundations. In fact, he has in the past sometimes written in a strongly anti–Christian vein, while the present book reflects the influence of, among others, Miri Rubin, whose Corpus Christi: The Eucharist in Late Medieval Culture is credited in a footnote. Steiner asserts that, after the Platonisms and Gnosticisms of late antiquity, it is the doctrines of incarnation and transubstantiation that mark ‘the disciplining of Western syntax and conceptualization’ in philosophy and art. ‘Every heading met with in a study of “creation,” every nuance of analytic and figural discourse,’ he says, derives from incarnation and transubstantiation, ‘concepts utterly alien to either Judaic or Hellenic perspectives– though they did, in a sense, arise from the collisions and commerce between both.’….
The incarnation of God in the Son, the transubstantiation of bread and wine into his body and blood, are ‘a mysterium, an articulated, subtly innervated attempt to reason the irrational at the very highest levels of intellectual pressure.’ ‘Uniquely, perhaps, the hammering out of the teaching of the eucharist compels Western thought to relate the depth of the unconscious and of pre-history with speculative abstractions at the boundaries of logic and of linguistic philosophy.’ Later, the ‘perhaps’ in that claim seems to have disappeared:
At every significant point, Western philosophies of art and Western poetics draw their secular idiom from the substratum of Christological debate. Like no other event in our mental history, the postulate of God’s kenosis through Jesus and of the never-ending availability of the Savior in the wafer and wine of the eucharist, conditions not only the development of Western art and rhetoric itself, but at a much deeper level, that of our understanding and reception of the truth of art– a truth antithetical to the condemnation of the fictive in Plato.
This truth reaches its unrepeated perfection in Dante, says Steiner. In Dante, ‘It rounds in glory the investigation of creativity and creation, of divine authorship and human poesis, of the concentric spheres of the aesthetic, the philosophical, and the theological. Now truth and fiction are made one, now imagination is prayer, and Plato’s exile of the poets refuted.’ In the fashionable critical theories of our day, we witness ‘endeavors of the aesthetic to flee from incarnation.’ ‘It is the old heresies which revive in the models of absence, of negation or erasure, of the deferral of meaning in late–twentieth–century deconstruction. The counter-semantics of the deconstructionist, his refusal to ascribe a stable significance to the sign, are moves familiar to [an earlier] negative theology.’ Heidegger’s poetics of ‘pure immanence’ are but one more attempt ‘to liberate our experience of sense and of form from the grip of the theophanic.’ But, Steiner suggests, attempted flights from the reality of Corpus Christi will not carry the day. ‘Two millennia are only a brief moment.’
Monday, January 23, 2006
Monday January 23, 2006
In Defense of Hilbert
(On His Birthday)
Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:
“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”
Harris adds in a footnote:
“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”
If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”
For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):
[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”
Some of Zach’s references:
[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].
[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990
From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:
“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”
[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….
[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.
[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.
From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:
“What Hilbert wants to secure is not the truth, but the consistency of the old analysis. This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics. In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one. The statements become meaningless figures built up from signs. Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess. Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs. One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess. And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae. By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game. The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules. Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color. Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game. Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge. What is meant to be recognized is that a contradiction can never occur as an end formula of a proof. Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration. One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color. In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game. It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together. Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”
Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:
“Hilbert’s program… received its share of criticism. Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’ But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time? I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous. It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)
Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:
“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”
One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136. The quotation in the English translation given there:
“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”
Corfield says of this quotation that he’d love to be told the original German. He should consult the above references cited by Richard Zach.
For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.
Wednesday, December 14, 2005
Wednesday December 14, 2005
to Eternity
For Loomis Dean
See also
For Rita Moreno
on Her Birthday
(Dec. 11, 2005)
Los Angeles Times
Tuesday, Dec. 13, 2005
OBITUARIES LOOMIS DEAN Loomis Dean, 88; By Jon Thurber, Times Staff Writer
Loomis Dean, a Life magazine photographer who made memorable pictures of the royalty of both Europe and Hollywood, has died. He was 88.
Dean died Wednesday [December 7, 2005] at Sonoma Valley Hospital in Sonoma, Calif., of complications from a stroke, according to his son, Christopher. In a photographic career spanning six decades, Dean's leading images included shirtless Hollywood mogul Darryl F. Zanuck trying a one-handed chin-up on a trapeze bar, the ocean liner Andrea Doria listing in the Atlantic and writer Ernest Hemingway in Spain the year before he committed suicide. One of his most memorable photographs for Life was of cosmopolitan British playwright and composer Noel Coward in the unlikely setting of the Nevada desert. Dean shot 52 covers for Life, either as a freelance photographer or during his two stretches as a staffer with the magazine, 1947-61 and 1966-69. After leaving the magazine, Dean found steady freelance work in magazines and as a still photographer on film sets, including several of the early James Bond movies starring Sean Connery. Born in Monticello, Fla., Dean was the son of a grocer and a schoolteacher. When the Dean family's business failed during the Depression, they moved to Sarasota, Fla., where Dean's father worked as a curator and guide at the John and Mable Ringling Museum of Art. Dean studied engineering at the University of Florida but became fascinated with photography after watching a friend develop film in a darkroom. He went off to what is now the Rochester Institute of Technology, which was known for its photography school. After earning his degree, Dean went to work for the Ringling circus as a junior press agent and, according to his son, cultivated a side job photographing Ringling's vast array of performers and workers. He worked briefly as one of Parade magazine's first photographers but left after receiving an Army Air Forces commission during World War II. During the war, he worked in aerial reconnaissance in the Pacific and was along on a number of air raids over Japan. His first assignment for Life in 1946 took him back to the circus: His photograph of clown Lou Jacobs with a giraffe looking over his shoulder made the magazine's cover and earned Dean a staff job. In the era before television, Life magazine photographers had some of the most glamorous work in journalism. Life assigned him to cover Hollywood. In 1954, the magazine published one of his most memorable photos, the shot of Coward dressed for a night on the town in New York but standing alone in the stark Nevada desert. Dean had the idea of asking Coward, who was then doing a summer engagement at the Desert Inn in Las Vegas, to pose in the desert to illustrate his song "Mad Dogs and Englishmen Go Out in the Midday Sun." As Dean recalled in an interview with John Loengard for the book "Life Photographers: What They Saw," Coward wasn't about to partake of the midday sun. "Oh, dear boy, I don't get up until 4 o'clock in the afternoon," Dean recalled him saying. But Dean pressed on anyway. As he related to Loengard, he rented a Cadillac limousine and filled the back seat with a tub loaded with liquor, tonic and ice cubes — and Coward. The temperature that day reached 119 as Coward relaxed in his underwear during the drive to a spot about 15 miles from Las Vegas. According to Dean, Coward's dresser helped him into his tuxedo, resulting in the image of the elegant Coward with a cigarette holder in his mouth against his shadow on the dry lake bed. "Splendid! Splendid! What an idea! If we only had a piano," Coward said of the shoot before hopping back in the car and stripping down to his underwear for the ride back to Las Vegas. In 1956, Life assigned Dean to Paris. While sailing to Europe on the Ile de France, he was awakened with the news that the Andrea Doria had collided with another liner, the Stockholm. The accident occurred close enough to Dean's liner that survivors were being brought aboard. His photographs of the shaken voyagers and the sinking Andrea Doria were some of the first on the accident published in a U.S. magazine. During his years in Europe, Dean photographed communist riots and fashion shows in Paris, royal weddings throughout Europe and noted authors including James Jones and William S. Burroughs. He spent three weeks with Hemingway in Spain in 1960 for an assignment on bullfighting. In 1989, Dean published "Hemingway's Spain," about his experiences with the great writer. In 1965, Dean won first prize in a Vatican photography contest for a picture of Pope Paul VI. The prize included an audience with the pope and $750. According to his son, it was Dean's favorite honor. In addition to his son, he is survived by a daughter, Deborah, and two grandsons. Instead of flowers, donations may be made to the American Child Photographer's Charity Guild (www.acpcg.com) or the Make-A-Wish Foundation. |
Related material:
The Big Time
(Log 24, July 29, 2003):
|
Saturday, December 10, 2005
Saturday December 10, 2005
Prequel on
Saint Cecilia’s Day
“Death itself would start
working backward.”
— Aslan in The Chronicles of Narnia
Celebrity Obits, Nov. 22, 2005 —
Intelligence and
Counterintelligence
(continued):
Aldous Huxley & C.S. Lewis both died on Nov.22, 1963. For some reason, their deaths went largely unnoticed… | The doors of perception lead to Narnia | November 22, 08:51:20am |
Shemp Howard died 50 years ago today | Moe | November 22, 09:17:18am |
See also the previous entry, and this follow-up:
“Shattuck’s death on Thursday… was reported by his nephew, John Shattuck, head of the John F. Kennedy Library Foundation, The Boston Globe reported Saturday.”
Related material:
“The White Witch rules Narnia,
and has brought to it
the Hundred Years of Winter.”
— The Narnia Academy
and the foundation of the
David Morrell Counterintelligence Library:
Shemp
Friday, December 2, 2005
Friday December 2, 2005
Proof 101
From a course description:
“This module aims to introduce the student to rigorous university level mathematics….
Syllabus: The idea of and need for mathematical statements and proofs…. proof by contradiction… proof by induction…. the infinite number of primes….”
In the December Notices of the American Mathematical Society, Brian (E. B.) Davies, a professor of mathematics at King’s College London, questions the consistency of Peano Arithmetic (PA), which has the following axioms:
From BookRags.com—
Axiom 1. 0 is a number.
Axiom 2. The successor of any number is a number.
Axiom 3. If a and b are numbers and if their successors are equal, then a and b are equal.
Axiom 4. 0 is not the successor of any number.
Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.
It should be noted that the word “number” as used in the Peano axioms means “non-negative integer.” The fifth axiom deserves special comment. It is the first formal statement of what we now call the “induction axiom” or “the principle of mathematical induction.”
Peano’s fifth axiom particularly troubles Davies, who writes elsewhere:
I contend that our understanding of number should be placed in an historical context, and that the number system is a human invention. Elementary arithmetic enables one to determine the number of primes less than twenty as certainly as anything we know. On the other hand Peano arithmetic is a formal system, and its internal consistency is not provable, except within set-theoretic contexts which essentially already assume it, in which case their consistency is also not provable. The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction, which is an abstraction from our everyday experience….
… Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated. Even today many university students who have been taught the principle of induction prefer to avoid its use, because they do not feel that it is as natural or as certain as a purely algebraic or geometric proof, if they can find one. The feelings of university students may not settle questions about what is truly fundamental, but they do give some insight into our native intuitions.— E. B. Davies in
“Counting in the real world,”
March 2003 (word format),
To appear in revised form in
Brit. J. Phil. Sci. as
“Some remarks on
the foundations
of quantum mechanics”
Exercise:
Discuss Davies’s claim that
The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction.
Cite the following passage in your discussion.
It will be clear by now that, if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate.
… I can hardly do better than go back to the Greeks. I will state and prove two of the famous theorems of Greek mathematics. They are “simple” theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class. Each is as fresh and significant as when it was discovered– two thousand years have not written a wrinkle on either of them. Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader, however slender his mathematical equipment.
I. The first is Euclid’s proof of the existence of an infinity of prime numbers.
The prime numbers or primes are the numbers
(A) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
which cannot be resolved into smaller factors. Thus 37 and 317 are prime. The primes are the material out of which all numbers are built up by multiplication: thus
666 = 2 . 3 . 3 . 37.
Every number which is not prime itself is divisible by at least one prime (usually, of course, by several). We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.
Let us suppose that it does, and that
2, 3, 5, . . . , P
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the numberQ = (2 . 3 . 5 . . . . . P) + 1.
It is plain that Q is not divisible by any of
2, 3, 5, …, P;
for it leaves the remainder 1 when divided by any one of these numbers. But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them. This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.
The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
— G. H. Hardy,
A Mathematician’s Apology,
quoted in the online guide for
Clear and Simple as the Truth:
Writing Classic Prose, by
Francis-Noël Thomas
and Mark Turner,
Princeton University Press
In discussing Davies’s claim that the above proof is by induction, you may want to refer to Davies’s statement that
Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated
and to Hardy’s statement that the above proof is due to Euclid.
Thursday, December 1, 2005
Thursday December 1, 2005
Campion’s Day
Today is the feast of the Catholic saint Edmund Campion. Campion, a Jesuit with a graceful prose style, would perhaps not be too deeply offended by the fact that his surname is now best known in some circles as that of a fictional character– the “Albert Campion” of Margery Allingham‘s detective stories.
The following is from a web page devoted to Allingham’s fiction, Roger Johnson’s “Thoughts on Mr. Campion and His Family.”
This is not of course to say that Albert Campion’s family are Roman Catholics: indeed, all the evidence is that they are Church of England. However, the fact does suggest a couple of interesting minor parallels with Mr Rudolph K–. St. Edmund Campion necessarily spent most of his mission in England incognito. And, perhaps more pertinently, the campion is a small hardy English wild flower – like the scarlet pimpernel, in fact. And the younger Albert Campion is strongly in the tradition of the apparently effete aristocrat Sir Percy Blakeney, who achieved fame as the dashing hero known as the Scarlet Pimpernel.
Richard Martin says that the name is “a barely disguised sign,” being the old French word for “champion.”
The non-fictional St. Edmund Campion is of course remembered also, for instance in the names of Campion College in suburban Sydney, of Campion College at the University of Regina and of Campion Hall at the University of Oxford.
Friday, November 11, 2005
Friday November 11, 2005
(continued)
Words Made Flesh: Code, Culture, Imagination—
… letters create things by the virtue of an algorithm…
— Sefer Yetzirah
Spelling is a sign, Elly. When you win the national bee, we'll know that you are ready to follow in Abulafia's footsteps. Once you're able to let the letters guide you through any word you are given, you will be ready to receive shefa."
In the quiet of the room, the sound of Eliza and her father breathing is everything.
"Do you mean," Eliza whispers, "that I'll be able to talk to God?"
Diamond Theory notes
of Feb. 4, 1986,
of April 26, 1986, and
of May 26, 1986,
Sacerdotal Jargon
(Log24, Dec. 5, 2002),
and 720 in the Book
(Log24, Epiphany 2004).
Tuesday, November 1, 2005
Tuesday November 1, 2005
The picture in the seal is of Plato's Academy.
"The great philosopher Plato excluded from his Academy anyone who had not studied geometry. He would have been delighted to admit Sir Michael Atiyah, who was for a time Savilian Professor of Geometry at Oxford…"
Would he?
Sir Michael Atiyah's
Anti-Platonism
"Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance….
I tend to think that science and mathematics are ways the human mind looks and experiences– you cannot divorce the human mind from it. Mathematics is part of the human mind. The question whether there is a reality independent of the human mind, has no meaning– at least, we cannot answer it."
— Sir Michael Atiyah, interview in Oslo, May 2004
"For Plato, the Forms represent truth, or reality…. these Forms are independent of the mind: they are eternal, unchanging and perfect."
— Roy Jackson (pdf)
Atiyah's denial of a reality independent of the human mind may have something to do with religion:
"Socrates and Plato were considered 'Christians before Christ'; they paved the way for the coming of Christianity by providing it with philosophical and theoretical foundations that would be acceptable to the western mind.
In the analogy of the cave, the sun represents the Form of the Good. In the same way that the sun is the source of all things and gives light to them, the Form of the Good is over and above the other Forms, giving them light and allowing us to perceive them. Therefore, when you have awareness of the Form of the Good you have achieved true enlightenment. In Christianity, the Form of the Good becomes God: the source of all things."
— Roy Jackson, The God of Philosophy (pdf)
Saturday, October 29, 2005
Saturday October 29, 2005
"I remember how the darkness doubled
I recall lightning struck itself
I was listening, listening to the rain
I was hearing, hearing something else
Life in the hive puckered up my night
The kiss of death, the embrace of life
There I stand neath the Marquee Moon
Just waiting"
— Tom Verlaine, "Marquee Moon"
In memory of Michael Gill,
producer and director of the
1969 TV series "Civilisation,"
who died on October 20:
Two descriptions of "Aeon Flux,"
a story featured in the Log24 entry
on the day that Gill died —
"After Aeon is done, Trevor decides that she knows too much, so he has a underling propose a plan to kill her. The plan, quite strangely, is to implant a bunch of nanites (microscopic robots) in Trevors seminal duct so he has sex with Aeon and the nanites tear her apart from the inside. But Aeon was prepared because she had some weird, mean, spiky, device in her uterus(!?!!) that eats the nanites (that part is kinda weak), she blows up a wall then and escapes leaving Trevor standing there naked and confused."
advocate of nanotechnology,
who died yesterday at 62:
The Incredible Shrinking Man
(Wired Magazine, October 2004)
See also yesterday's entry on Scientism.
In memory of
Thomas Wootton Masland,
brother of
Richard Harry Masland, Harvard '64,
the Log24 entries of October 25.
Thursday, August 25, 2005
Thursday August 25, 2005
Train of Thought
Part I: The 24-Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota |
Like footprints erased in the sand….
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that space-time should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete Space-Time
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete space-time structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.
in a Discrete Space
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Tuesday, March 15, 2005
Tuesday March 15, 2005
Religion at Harvard
The Children’s Hour
Harvard Magazine,
Sept.-Oct. 2004:
“With the legalization of gay marriage in Massachusetts, Harvard couples were among those who took vows…. Lowell House master Diana Eck (left) and co-master Dorothy Austin tied the knot in Memorial Church on July 4, with Rev. Peter Gomes, Plummer professor of Christian morals, officiating.”
Once in Love with Amy
“In the Dining Hall are portraits of President Lowell and his wife; his sister Amy Lowell (Pulitzer prize winning poet, and a lover of scandal…)….” |
Today’s Harvard Crimson: “Stone joined members of the Foundation for lunch yesterday in Lowell House before delivering her remarks at Memorial Church last night…” |
Hold That Thought
nothing – the word had sexual connotations, as a slang word referring to female sexual parts. Compare Hamlet:
HAMLET Lady, shall I lie in your lap?
[Lying down at OPHELIA’s feet]
OPHELIA No, my lord.
HAMLET I mean, my head upon your lap?
OPHELIA Ay, my lord.
HAMLET Do you think I meant country matters?
OPHELIA I think nothing, my lord.
HAMLET That’s a fair thought to lie between maids’ legs.
OPHELIA What is, my lord?
HAMLET Nothing.
— Hamlet, III.2
Wednesday, December 29, 2004
Wednesday December 29, 2004
The Dark Door
From Log24.net, Dec. 22, 2003:
“One, two, three, and we began to sing, our voices high and seemingly distant in the snow-felted darkness round the house that was occupied by nobody we knew. We stood close together, near the dark door.
Good King Wenceslas looked out
On the Feast of Stephen.”
— Dylan Thomas,
A Child’s Christmas in Wales
“The day after Christmas
turned out to be a living nightmare.”
— Arthur C. Clarke, Dec. 27, 2004
Adapted from the logo of the
Arthur C. Clarke Foundation:
Dabo claves regni caelorum. By silent shore
Ripples spread from castle rock. The metaphor
For metamorphosis no keys unlock.
— “Endgame,” Steven H. Cullinane,
November 7, 1986
Saturday, September 18, 2004
Saturday September 18, 2004
The First Idea
From aldaily.com,
a service of
The Chronicle of
Higher Education:
New Books“Mr. Oppenheimer, given what has happened since, would you again accept to develop the bomb? Even after Hiroshima?” “Yes”… more» The foundations of civilization are but modest: consider for instance games of peekaboo and patty-cake… more» |
Peekaboo:
Wallace Stevens on “The First Idea”
Patty-cake:
“A specialist in Homer’s Odyssey and early Greek lyric poetry, Joseph Russo is the only American classicist among six international scholars to provide commentary for Oxford University Press’ three-volume edition of the epic poem.”
— Introduction to an inaugural lecture, “Language, Poetry, Philology, and ‘The Stateliest Measure,'” at Haverford College given by Joseph Russo on Feb. 26, 1999
Sunday, August 29, 2004
Sunday August 29, 2004
Olympic Arc
Thomas Becker, president of Chautauqua Institution, on Friday, Aug. 27, 2004:
“I’m really proud of this lecture platform this year. We started with Phil Wilcox on the first day of the season and finished with Sandra Day O’Connor. The arc of participation between them was really amazing.”
Phil Wilcox: See
Israel and Palestine:
Let’s Separate Myth from Reality,
by Philip C. Wilcox, Jr., President,
Foundation for Middle East Peace,
Chautauquan Daily, June 28, 2004
Sandra Day O’Connor: See
The Majesty of the Law:
Reflections of a
Supreme Court Justice,
by Sandra Day O’Connor
The O’Connor link above is to a page at the Chautauqua Bookstore.
For Justice O’Connor:
Reflections on Themis
(Log24, Aug. 17, 2004)
For Wilcox:
The Zen of Abraham
(Same entry, different title.)
I personally was at Chautauqua only one day this season — Friday, the 13th of August. My stops of course included the Chautauqua Bookstore, where I purchased the following:
Human cultural activity is mostly what Walker Percy astutely called “symbol-mongering.” Of the three books above, the central one offers the best symbols.
My own version of a
Chautauqua “Versus” symbol:
For further details, see the
For an “arc” symbol, see
Saturday, June 5, 2004
Saturday June 5, 2004
continued…
of the 3×3 square
are discussed in
From Lullus to Cognitive Semantics:
The Evolution of a Theory of Semantic Fields
by Wolfgang Wildgen and in
Another Page in the Foundation of Semiotics:
A Book Review of On the Composition of Images, Signs & Ideas, by Giordano Bruno…
by Mihai Nadin
“We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn’t merely sensational, that doesn’t get its message across in 10 seconds, that isn’t falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game.”
— Robert Hughes, speech of June 2, 2004
Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder.
For a meditation on the related 4×4 square grid as “art that holds time,” see Time Fold.
Tuesday, April 6, 2004
Tuesday April 6, 2004
Ideas and Art, Part III
The first idea was not our own. Adam
In Eden was the father of Descartes…
— Wallace Stevens, from
Notes Toward a Supreme Fiction
"Quaedam ex his tanquam rerum imagines sunt, quibus solis proprie convenit ideae nomen: ut cùm hominem, vel Chimaeram, vel Coelum, vel Angelum, vel Deum cogito."
— Descartes, Meditationes III, 5
"Of my thoughts some are, as it were, images of things, and to these alone properly belongs the name idea; as when I think [represent to my mind] a man, a chimera, the sky, an angel or God."
— Descartes, Meditations III, 5
Begin, ephebe, by perceiving the idea
Of this invention, this invented world,
The inconceivable idea of the sun.
You must become an ignorant man again
And see the sun again with an ignorant eye
And see it clearly in the idea of it.
— Wallace Stevens, from
Notes Toward a Supreme Fiction
"… Quinimo in multis saepe magnum discrimen videor deprehendisse: ut, exempli causâ, duas diversas solis ideas apud me invenio, unam tanquam a sensibus haustam, & quae maxime inter illas quas adventitias existimo est recensenda, per quam mihi valde parvus apparet, aliam verò ex rationibus Astronomiae desumptam, hoc est ex notionibus quibusdam mihi innatis elicitam, vel quocumque alio modo a me factam, per quam aliquoties major quàm terra exhibetur; utraque profecto similis eidem soli extra me existenti esse non potest, & ratio persuadet illam ei maxime esse dissimilem, quae quàm proxime ab ipso videtur emanasse."
— Descartes, Meditationes III, 11
"… I have observed, in a number of instances, that there was a great difference between the object and its idea. Thus, for example, I find in my mind two wholly diverse ideas of the sun; the one, by which it appears to me extremely small draws its origin from the senses, and should be placed in the class of adventitious ideas; the other, by which it seems to be many times larger than the whole earth, is taken up on astronomical grounds, that is, elicited from certain notions born with me, or is framed by myself in some other manner. These two ideas cannot certainly both resemble the same sun; and reason teaches me that the one which seems to have immediately emanated from it is the most unlike."
— Descartes, Meditations III, 11
"Et quamvis forte una idea ex aliâ nasci possit, non tamen hîc datur progressus in infinitum, sed tandem ad aliquam primam debet deveniri, cujus causa sit in star archetypi, in quo omnis realitas formaliter contineatur, quae est in ideâ tantùm objective."
— Descartes, Meditationes III, 15
"And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality [or perfection] that is found objectively [or by representation] in these ideas is contained formally [and in act]."
— Descartes, Meditations III, 15
Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction,"
The Quest for the Fiction of the Absolute:
"Canto nine considers the movement of the poem between the particular and the general, the immanent and the transcendent: "The poem goes from the poet's gibberish to / The gibberish of the vulgate and back again. / Does it move to and fro or is it of both / At once?" The poet, the creator-figure, the shadowy god-figure, is elided, evading us, "as in a senseless element." The poet seeks to find the transcendent in the immanent, the general in the particular, trying "by a peculiar speech to speak / The peculiar potency of the general." In playing on the senses of "peculiar" as particular and strange or uncanny, these lines play on the mystical relation of one and many, of concrete and abstract."
Brian Cronin in Foundations of Philosophy:
"The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight."
— From Ch. 2, Identifying Direct Insights
Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction":
"The fourth canto returns to the theme of opposites. 'Two things of opposite natures seem to depend / On one another . . . . / This is the origin of change.' Change resulting from a meeting of opposities is at the root of Taoism: 'Tao produced the One. / The One produced the two. / The two produced the three. / And the three produced the ten thousand things' (Tao Te Ching 42) …."
From an entry of March 7, 2004:
From the web page
Introduction to the I Ching– "He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the 'great primal beginning' of all that exists, t'ai chi — in its original meaning, the 'ridgepole.' Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin, . This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites." The t'ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that
"W.J.T. Mitchell calls 'iconology' A variation on the t'ai chi symbol appears in a log24.net entry for March 5:
The Line, See too my web page Logos and Logic, which has the following:
Logos Alogos, In the conclusion of Section 3, Canto X, of "Notes," Stevens says
This is the logoi alogoi of Simone Weil. |
In "Notes toward a Supreme Fiction,"
Wallace Stevens lists three criteria
for a work of the imagination:
It Must Be Abstract
The Line,
by S.H. Cullinane
It Must Change
It Must Give Pleasure
Related material:
Tuesday April 6, 2004
Ideas and Art, Part II
"We do not, of course, see ideas."
— Roger Kimball, Minimalist Fantasies, 2003
"Idea (Lat. idea , forma , species ; Gk. idea , eidos , from idein , to see; Fr. idée ; Ger. Bild ; Begriff )
Probably to no other philosophical term have there been attached so many different shades of meaning as to the word idea. Yet what this word signifies is of much importance. Its sense in the minds of some philosophers is the key to their entire system. But from Descartes onwards usage has become confused and inconstant. Locke, in particular, ruined the term altogether in English philosophical literature…."
— The Catholic Encylopedia, 1910
James Hillman, A Blue Fire , p. 53:
"For us ideas are ways of regarding things (modi res considerandi ), perspectives. Ideas give us eyes, let us see …. Ideas are ways of seeing and knowing….
Our word idea comes from the Greek eidos , which meant originally in early Greek thought, and as Plato used it, both that which one sees — an appearance or shape in a concrete sense — and that by means of which one sees. We see them, and by means of them. Ideas are both the shape of events, their constellation in this or that archetypal pattern, and the modes that make possible our ability to see through events into their pattern. By means of an idea we can see the idea cloaked in the passing parade. The implicit connection between having ideas to see with and seeing ideas themselves suggests that the more ideas we have, the more we see, and the deeper the ideas we have, the deeper we see. It also suggests that ideas engender other ideas, breeding new perspectives for viewing ourselves and world.
Moreover, without them we cannot 'see' even what we sense with the eyes in our heads, for our perceptions are shaped according to particular ideas …. And our ideas change as changes take place in the soul, for as Plato said, soul and idea refer to each other, in that an idea is the 'eye of the soul,' opening us through its insight and vision."
Hillman does not say where in Plato this extraordinary saying, that an idea is the eye of the soul, occurs. He is probably wrong.
Both Kimball and Hillman seem confused.
A more sensible approach to these matters is available in Brian Cronin's Foundations of Philosophy:
"3.4 An Insight Pivots between the Abstract and the Concrete
On the one hand, an insight is dealing with data and images which are concrete and particular: Archimedes had one chalice, one King, and one particular problem to solve. On the other hand, what the insight grasps is an idea, a relation, a universal, a law; and that is abstract. The laws that Archimedes eventually formulated were universal, referring not only to this chalice but also to any other material body immersed in any other liquid at any time or any place. The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight. First let us now clarify the difference between images, ideas and concepts…."
Wednesday, January 14, 2004
Wednesday January 14, 2004
Games
On this date —
Alfred Tarski was born
in 1902 in Warsaw, and
Kurt Friedrich Gödel died
in 1978 in Princeton.
From last year’s entry on this date:
What is Truth?
“What is called ‘losing’ in chess
may constitute winning
in another game.”
— Ludwig Wittgenstein,
Remarks on the
Foundations of Mathematics
(revised edition, MIT Press, 1978)
Tuesday, September 2, 2003
Tuesday September 2, 2003
One Ring to Rule Them All
In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.
Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:
“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’ Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”
— Leonard Gillman: An Interview
This clears up a question I asked earlier in this journal….
Wednesday, May 14, 2003 Common Sense On the mathematician Kolmogorov: “It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.” — Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003 That this theorem is Kolmogorov’s is news to me. See
The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses. Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case. A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15. This is cited by Gillman and Jerison in the classic Rings of Continuous Functions. There ARE some references that indicate Kolmogorov may have done some work of his own in this area. See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”). Any other references to Kolmogorov’s work in this area would be of interest. Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone. I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968. |
A response by Richard Cudney:
“In regard to your entry, it is largely correct. The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works. So, I suppose my entry was unfair to Gelfand. You’re right, the issue of credit is a bit touchy since Gelfand was his student. In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student. At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst. In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A. That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up. This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras. In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique. That is to say, this whole vein of Gelfand’s work started in this joint paper. Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge. Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third). As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin. The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.” |
Friday, July 25, 2003
Thursday, May 22, 2003
Thursday May 22, 2003
Mental Health Month:
Springtime for Wagner
“And now what you’ve all been waiting for…
— Colin Hay as Zac in the film “Cosi“
“When I sought those who would sympathize with my plans, I had only you, the friends of my particular art, my most personal work and creation, to turn to.”
— Wagner’s address at the ceremony for the laying of the foundation stone of the Festival Theater in Bayreuth, May 22 (Wagner’s birthday), 1872
“The new computer package DISCRETA which was created in Bayreuth is in the process of permanent development.”
— “A Computer Approach to the Enumeration of Block Designs Which Are Invariant With Respect to a Prescribed Permutation Group”
The above is a preprint from Dresden.
See, too, the work of Bierbrauer, who received his doctorate at Mainz in 1977 and taught at Heidelberg from 1977 to 1994. Bierbrauer’s lecture notes give a particularly good background for the concepts involved in my Diamond Theory, in the tradition of Witt and Artin. See
Introduction to Group Theory
and Applications,
by Jürgen Bierbrauer, 138 pp., PostScript
Thursday, April 24, 2003
Thursday April 24, 2003
Cross-Referenced
†
Shortly after midnight on the night of April 22-23, I updated my entry for Shakespeare's birthday with the following quotation:
"With a little effort, anything can be shown to connect with anything else: existence is infinitely cross-referenced."
— Opening sentence of Martha Cooley's The Archivist
About 24 hours later, I came across the following obituary in The New York Times:
"Edgar F. Codd, a mathematician and computer scientist who laid the theoretical foundation for relational databases, the standard method by which information is organized in and retrieved from computers, died on Friday…. He was 79."
The Times does not mention that the Friday it refers to is Good Friday. God will have his little jokes.
From Computerworld.com: |
|||
---|---|---|---|
|
1969: Edgar F. “Ted” Codd invents the relational database. 1973: Cullinane, led by John J. Cullinane, ships IDMS, a network-model database for IBM mainframes. 1976: Honeywell ships Multics Relational Data Store, the first commercial relational database. |
For a better (and earlier) obituary than the Times's, see The San Jose Mercury News of Easter Sunday. For some thoughts on death and the afterlife appropriate to last weekend, see The Matthias Defense.
† The Exorcist, 1973
Wednesday, March 12, 2003
Wednesday March 12, 2003
Daimon Theory
Today is allegedly the anniversary of the canonization, in 1622, of two rather important members of the Society of Jesus (Jesuits):
Ignatius Loyola…
Click here for Loyola’s legacy of strategic intelligence.
Francis Xavier…
Click here for Xavier’s legacy of strategic stupidity.
We can thank (or blame) a Jesuit (Gerard Manley Hopkins) for the poetic phrase “immortal diamond.” He may have been influenced by Plato, who has Socrates using a diamond figure in an argument for the immortality of the soul. Confusingly, Socrates also talked about his “daimon” (pronounced dye-moan). Combining these similar-sounding concepts, we have Doctor Stephen A. Diamond writing about daimons — a choice of author and topic that neatly combines the strategic intelligence of Loyola with the strategic stupidity of Xavier.
The cover illustration is perhaps not of Dr. Diamond himself.
A link between diamond theory and daimon theory is furnished by the charitable legacy of the non-practicing Jew Walter Annenberg.
For Annenberg and diamond theory, see this site on the elementary geometry of quilt blocks, which credits the Annenberg Foundation for support.
For Annenberg and daimon theory, see this site on Socrates, which has a similar Annenberg support credit.
Advanced disciples of Annenberg can learn much from the Perseus site about daimon theory. Let us pray that Abrahamic religious bigotry does not stand in their way. Less advanced disciples of Annenberg may find fulfillment in teaching children the beauty of elementary 4×4 quilt-block symmetry. Let us pray that academic bigotry does not prevent these same children, when they have grown older, from learning the deeper, and more difficult, beauties of diamond theory.
Tuesday, January 14, 2003
Tuesday January 14, 2003
Remarks on Day 14 of
the Year of Our Lord 2003
On this date —
Alfred Tarski was born in 1902 in Warsaw, and
Kurt Friedrich Gödel died in 1978 in Princeton.
What is Truth?
“What is called ‘losing’ in chess may constitute winning in another game.”
Cited in “A Note on Wittgenstein’s ‘Notorious Paragraph’ about the Gödel Theorem,” by Juliet Floyd (Boston University) and Hilary Putnam (Harvard University), Journal of Philosophy (November 2000), 45 (11): 624-632.
See also
Juliet Floyd’s “Prose versus proof : Wittgenstein on Gödel, Tarski and truth,” Philosophia Mathematica 3, vol. 9 (2001): 901-928,
and
Juliet Floyd’s “The Rule of the Mathematical: Wittgenstein’s Later Discussions.” PhD Dissertation, Harvard University, 1990. Abstract in Dissertation Abstracts International (June 1991), 51 (12A): 4146-A:
“My thesis aims to defend Wittgenstein from the charges of benighted arrogance traditionally levelled against him.”
Romeo: O, she doth teach
the torches to burn bright!
— “Romeo and Juliet,” Act One, Scene V
Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (revised edition, Cambridge, Mass.: MIT Press, 1978)
Saturday, January 11, 2003
Saturday January 11, 2003
METROPOLITAN ART WARS:
The First Days of Disco
Some cultural milestones, in the order I encountered them today:
From Dr. Mac’s Cultural Calendar:
- “On this day in 1963, Whiskey-A-Go-Go—believed to be the first discotheque in the world—opened on Sunset Boulevard in Los Angeles with extraordinary hype and fanfare.”
From websites on Whit Stillman’s film, “The Last Days of Disco”:
Scene: Manhattan in the very early 1980’s.
Alice and her friend Charlotte are regulars at a fashionable disco.
“Charlotte is forever giving poor Alice advice about what to say and how to behave; she says guys like it when a girl uses the word ‘sexy,’ and a few nights later, when a guy tells Alice he collects first editions of Scrooge McDuck comic books, she…”
“… looks deep into his eyes and purrs ‘I think Scrooge McDuck is sexy!’ It is a laugh-out-loud funny line and a shrewd parody, but is also an honest statement.”
(Actually, to be honest, I encountered Thomson first and Ebert later, but the narrative sequence demands that they be rearranged.)
The combination of these cultural landmarks suggested that I find out what Scrooge McDuck was doing during the first days of disco, in January 1963. Some research revealed that in issue #40 of “Uncle Scrooge,” with a publication date of January 1963, was a tale titled “Oddball Odyssey.” Plot summary: “A whisper of treasure draws Scrooge to Circe.”
Further research produced an illustration:
Desiring more literary depth, I sought more information on the story of Scrooge and Circe. It turns out that this was only one of a series of encounters between Scrooge and a character called Magica de Spell. The following is from a website titled
“Magica’s first appearance is in ‘The Midas Touch’ (US 36-01). She enters the Money Bin to buy a dime from Scrooge. Donald tells Scrooge that she is a sorceress, but Scrooge sells her a dime anyway. He sells her his first dime by accident, but gets it back. The fun starts when Scrooge tells her that it is the first dime he earned. She is going to make an amulet….”
with it. Her pursuit of the dime apparently lasts through a number of Scrooge episodes.
“…in Oddball Odyssey (US 40-02). Magica discovers Circe’s secret cave. Inside the cave is a magic wand that she uses to transform Huey, Dewey and Louie to pigs, Donald to a goat (later to a tortoise), and Scrooge to a donkey. This reminds us of the treatment Circe gave Ulysses and his men. Magica does not succeed in transforming Scrooge after stealing the Dime, and Scrooge manages to break the spell (de Spell) by smashing the magic wand.”
At this point I was reminded of the legendary (but true) appearance of Wallace Stevens’s wife on another historic dime. This was discussed by Charles Schulz in a cartoon of Sunday, May 27, 1990:
Here Sally is saying…
Who, me?… Yes, Ma’am, right here.
This is my report on dimes and pennies…
“Wallace Stevens was a famous poet…
His wife was named Elsie…”
“Most people do not know that Elsie was the model for the 1916 ‘Liberty Head’ dime.”
“Most people also don’t know that if I had a dime for every one of these stupid reports I’ve written, I’d be a rich person.”
Finally, sitting outside the principal’s office:
I never got to the part about who posed for the Lincoln penny.
I conclude this report on a note of synchronicity:
The above research was suggested in part by a New York Times article on Ovid’s Metamorphoses I read last night. After locating the Scrooge and Stevens items above, I went to the Times site this afternoon to remind myself of this article. At that point synchronicity kicked in; I encountered the following obituary of a Scrooge figure from 1963… the first days of disco:
The New York Times, January 12, 2003 (So dated at the website on Jan. 11) C. Douglas Dillon Dies at 93;
|
Et cetera, et cetera, and so forth.
(See yesterday’s two entries, “Something Wonderful,” and “Story.”)
Two reflections suggest themselves:
“I need a photo opportunity.
I want a shot at redemption.
Don’t want to end up a cartoon
In a cartoon graveyard.”
— Paul Simon
Ending up in a cartoon graveyard is indeed an unhappy fate; on the other hand…
It is nice to be called “sexy.”
Added at 1:50 AM Jan. 12, 2003:
Tonight’s site music, in honor of Mr. Dillon
and of Hepburn, Holden, and Bogart in “Sabrina” —
“Isn’t It Romantic?”
Monday, October 21, 2002
Monday October 21, 2002
Birthdays for a Small Planet
Today's birthdays:
- Ursula K. Le Guin, 1929
- Frances FitzGerald, 1940
- Carrie Fisher, 1956
The entry below, "Theology for a Small Planet," sketches an issue that society has failed to address since the fall of 1989, when it was first raised by the Harvard Divinity Bulletin.
In honor mainly of Ursula K. Le Guin, but also of her fellow authors above, I offer Le Guin's solution. It is not new. It has been ignored mainly because of the sort of hateful and contemptible arrogance shown by
- executives in the tradition of Henry Ford and later Ford Foundation and Ford Motors employees McGeorge Bundy and Robert McNamara (see yesterday's entry below for Ford himself), by
- theologians in the tradition of the Semitic religions — Judaism, Christianity, and Islam — and by
- self-proclaimed "shamans of scientism" like Michael Shermer in the tradition of Scientific American magazine.
Here is an introduction to the theology that should replace the ridiculous and outdated Semitic religions.
"Scholarly translators of the Tao Te Ching, as a manual for rulers, use a vocabulary that emphasizes the uniqueness of the Taoist 'sage,' his masculinity, his authority. This language is perpetuated, and degraded, in most popular versions. I wanted a Book of the Way accessible to a present-day, unwise, unpowerful, and perhaps unmale reader, not seeking esoteric secrets, but listening for a voice that speaks to the soul. I would like that reader to see why people have loved the book for 2500 years.
It is the most lovable of all the great religious texts, funny, keen, kind, modest, indestructibly outrageous and inexhaustibly refreshing. Of all the deep springs, this is the purest water. To me it is also the deepest spring."
Tao Te Ching: Chapter 6
translated by Ursula K. Le Guin
The valley spirit never dies
Call it the mystery, the woman.
The mystery,
the Door of the Woman,
is the root
of earth and heaven.
Forever this endures, forever.
And all its uses are easy.
Wednesday, October 9, 2002
Wednesday October 9, 2002
ART WARS:
Apollo and Dionysus
From the New York Times of October 9, 2002:
Daniel Deverell Perry, a Long Island architect who created the marble temple of art housing the Sterling and Francine Clark Art Institute in Williamstown, Mass., died Oct. 2 in Woodstock, N.Y…. He was 97.
From The Birth of Tragedy, by Friedrich Nietzsche (tr. by Shaun Whiteside):
Chapter 1….
To the two gods of art, Apollo and Dionysus, we owe our recognition that in the Greek world there is a tremendous opposition, as regards both origins and aims, between the Apolline art of the sculptor and the non-visual, Dionysiac art of music.
Chapter 25….
From the foundation of all existence, the Dionysiac substratum of the world, no more can enter the consciousness of the human individual than can be overcome once more by that Apolline power of transfiguration, so that both of these artistic impulses are forced to unfold in strict proportion to one another, according to the law of eternal justice. Where the Dionysiac powers have risen as impetuously as we now experience them, Apollo, enveloped in a cloud, must also have descended to us; some future generation will behold his most luxuriant effects of beauty.
Notes:
- On the Clark Art Institute, from Perry’s obituary in the Times:
“When it opened in 1955, overlooking 140 acres of fields and ponds, Arts News celebrated its elegant galleries as the ‘best organized and most highly functional museum erected anywhere.'”
- The “Nymphs and Satyr” illustration above is on the cover of “CAI: Journal of the Clark Art Institute,” Volume 3, 2002. It is a detail from the larger work of the same title by William Bouguereau.
- Today, October 9, is the anniversary of the dedication in 28 B.C. of the Temple to Apollo on the Palatine Hill in Rome. See the journal entry below, which emphasizes the point that Apollo and Dionysus are not as greatly opposed as one might think.
Thursday, September 19, 2002
Thursday September 19, 2002
William Golding
and the Lost Boys
Author William Golding was born on this date in 1911.
Theater review,
‘The Terrible Tragedy of Peter Pan’
at House Theatre in ChicagoBy Chris Jones
“J. M. Barrie’s famous 1904 tale of Peter Pan and the Lost Boys is fertile ground for post-modern exploration.”
See also the Stephen King novel
(Forget the movie, which does not even mention William Golding.)
For a somewhat more cheerful variation on the Lost Boys theme, see the new
Kingdom Hearts game.
Of course, mature audiences might react to this Disney production by recalling the classic question, “Why did Mickey Mouse divorce Minnie Mouse?”
See also the
Lord of the Flies game
at the Nobel Prize Foundation site.
Monday, September 16, 2002
Monday September 16, 2002
A Time to Gather Stones Together
(Ecclesiastes 3:5)
Readings for Yom Kippur:
- “Stones,” from Jewish Heritage Online Magazine
- “The Foundation Stone: The Center of Creation,” from Jewish Heritage Online Magazine
- Many Dimensions, a novel by Charles Williams
Friday, August 2, 2002
Friday August 2, 2002
Double Day… August 2, 2002
“Time cannot exist without a soul (to count it).” — Aristotle
The above quotation appears in my journal note of August 2, 1995, as an epigraph on the reproduced title page of The Sense of an Ending, by Frank Kermode (Oxford University Press, 1967).
August 2, 1995, was the fortieth anniversary of Wallace Stevens’s death. On the same date in 1932 — seventy years ago today — actor Peter O’Toole was born. O’Toole’s name appears, in a suitably regal fashion, in my journal note of August 2, 1995, next to the heraldic crest of Oxford University, which states that “Dominus illuminatio mea.” Both the crest and the name appear below the reproduced title page of Kermode’s book — forming, as it were, a foundation for what Harvard professor Marjorie Garber scornfully called “the Church of St. Frank” (letters to the editor, New York Times Book Review, July 30, 1995).
Meditations for today, August 2, 2002:
From page 60 of Why I Am a Catholic, by Gary Wills (Houghton Mifflin, 2002):
“Was Jesus teasing Peter when he called him ‘Rocky,’ naming him ab opposito, as when one calls a not-so-bright person Einstein?”
From page 87 of The Third Word War, by Ian Lee (A&W Publishers, Inc., New York, 1978):
“Two birds… One stone (EIN STEIN).”
From “Seventy Years Later,” Section I of “The Rock,” a poem by Wallace Stevens:
A theorem proposed between the two —
Two figures in a nature of the sun….
From page 117 of The Sense of an Ending:
“A great many different kinds of writing are called avant-garde…. The work of William Burroughs, for instance, is avant-garde. His is the literature of withdrawal, and his interpreters speak of his hatred for life, his junk nihilism, his treatment of the body as a corpse full of cravings. The language of his books is the language of an ending world, its aim… ‘self-abolition.'”
From “Today in History,” by The Associated Press:
“Five years ago: ‘Naked Lunch’ author William S. Burroughs, the godfather of the ‘Beat generation,’ died in Kansas City, Mo., at age 83.”
Part of the above statement is the usual sort of AP disinformation, due not to any sinister intent but to stupidity and carelessness. Burroughs actually died in Lawrence, Kansas. For the location of Lawrence, click on the link below. Location matters.
From page 118 of The Sense of an Ending:
“Somewhere, then, the avant-garde language must always rejoin the vernacular.”
From the Billie Holiday songbook:
“Good mornin’, heartache.”
From page 63 of The New Yorker issue dated August 5, 2002:
“Birthday, death-day — what day is not both?” — John Updike
October 11th, 2007 at 1:38 am
once we reach the point at which the templeton foundation – or any other private sponsor for that matter – is the main source of funding in a certain area of science it would be time for society to react. react by outdoing the private source and thus claiming the research topic in question firmly back into the public domain.
if society chooses to be oblivious – well – then so be it. research in that area will then not be driven by public interest but by private interest. ultimately it is just a reflection of the value commonly assigned to a specific field.
what i hope this will ultimately achieve is to ring the alarm bell in society that no private organization should take over research funding and direction.
if this will not happen – well – then we are kind of lost anyways. and funding no matter what agenda behind is still better than no funding, since i firmly believe that ultimately the truth (i.e. true statements about reproducible empirical relations) will ultimately prevail and nothing else.
October 11th, 2007 at 3:53 pm
Chris says the truth consists of “true statements about reproducible empirical relations.” He should read William Golding’s Nobel lecture: “When I consider a universe which the scientist constructs by a set of rules which stipulate that this construct must be repeatable and identical, then I am a pessimist and bow down before the great god Entropy. I am optimistic when I consider the spiritual dimension which the scientist’s discipline forces him to ignore.”