A recent book on mathematics and art
from Princeton University Press, with a
foreword by Neil deGrasse Tyson —
Not to put too fine a point on it —
From an earlier post —
Tuesday, May 9, 2017
Art Space at Princeton
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Friday, September 21, 2018
Friday, July 20, 2018
Monday, October 24, 2016
Princeton Space
|
From the Thursday, Oct. 13, 2016, Daily Princetonian — The ever-elusive “space” is a word spoken into a great expanse of hopes and fears and delusions: “safe spaces,” “inclusive spaces,” “open spaces,” “green spaces,” “learning spaces.” In this space, words float around abstractly, almost effortlessly, seemingly without the weight of any gravity; appearing to be a distant glimmer of an idea, a once bright and assuring light, which— without much definition— easily fades into obscurity. Coming to Princeton, it’s tempting to feel as though the rhetoric surrounding the term “space” stretches the word out, magnifies it, and tacks it onto well-designed brochures and anonymous invitations. Yet the question remains— how do you comfortably situate yourself within the incredibly abstruse concept of “space,” especially when you happen to exist in a territory that has been occupied and claimed by an endless sea of others, and which has been upheld by an impregnable and deeply rooted history? In the process of interviewing various members of the University, one thing has become clear; the question of space is an issue that is pertinent to all members of the Princeton community. |
For greater depth on this topic, see the previous post.
For less depth, see a post of January 18, 2005.
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Monday, April 11, 2016
Like Decorations in a Cartoon Graveyard
From Sunday evening's In Memoriam post —
The "from Princeton" remark in the previous post came from
Princeton, but originated with a retired professor in Rochester,
NY, one Joseph Neisendorfer.
Another remark by Neisendorfer, from his weblog —
Those familiar with the chapter on Galois in the
Eric Temple Bell classic Men of Mathematics
will know that the words quoted above by
Neisendorfer are definitely not those of Albert Einstein.
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Tuesday, November 10, 2015
Princeton Symmetry
From this journal nine years ago today, on the
anniversary of Stanley finding Livingstone —
Click on the image for the Princeton connection.
Related art — Search Log24 for Time + Eternity.
See as well the theater producer pictured in last night's post
and a Princeton-related* review of one of his productions.
Footnote of November 11, 2015:
* Related, that is, only by the "Princeton connection" mentioned above.
For another Princeton connection of interest, see a symposium at
Princeton University on May Day, 2015 —
THE PEDAGOGY OF IMAGES:
DEPICTING COMMUNISM FOR CHILDREN
A sample symposium participant:
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Thursday, November 5, 2015
ABC Art or: Guitart Solo
“… the A B C of being….” — Wallace Stevens
Scholia —
Compare to my own later note, from March 4, 2010 —
“It seems that Guitart discovered these ‘A, B, C’ generators first,
though he did not display them in their natural setting,
the eightfold cube.” — Borromean Generators (Log24, Oct. 19)
See also Raiders of the Lost Crucible (Halloween 2015)
and “Guitar Solo” from the 2015 CMA Awards on ABC.
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Thursday, August 27, 2015
The Space of Art
"Thinking Outside the Square:
Support for Landscape and Portrait
Formats on Instagram"
Related material from March 18, 2015 —
Play Is Not Playing Around
|
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Tuesday, June 9, 2015
Princeton Flashback
From The Daily Princetonian on May 29, 2015:
"… well, isn’t that what Reunions is all about?
Making memories?"
"Try to remember the kind of September …."
From this journal on May 29, 2015:
Openings
|
The Dark Horse Rises
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Monday, April 6, 2015
Springtime for Princeton*
… Continues.
* And the late Julie Wilson.
For the title, see a Log24 search.
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Thursday, January 15, 2015
Princeton Music continues…
A post yesterday linked to a discussion
of the Faustian music of Milton Babbitt,
a serial composer who reportedly died
in Princeton on Saturday, Jan. 29, 2011.
Related material from this journal in
January 2005:
See also "me into you, you into me"
("Taking Lucifer Seriously," Jan. 24, 2004)
and the Saturday night "cold open" in this
journal on the date of Babbitt's death.
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Tuesday, January 13, 2015
Princeton Requiem
Angel Eyes
|
From The Daily Princetonian ,
U. community gathers to BY JACOB DONNELLY Students, faculty, staff and community members circled around a table supporting a single lit candle in the lobby of Murray-Dodge Hall on Monday night as they remembered the life of Audrey Dantzlerward ’16, who was found dead in her room in Edwards Hall today. The gathering, led by Dean of Religious Life and the Chapel Alison Boden, was moved to the lobby after a room reserved for the meeting overflowed.
Participants spoke commonly of Dantzlerward’s contributions to campus life, sharp intellect, supportive gestures and friendly demeanor, and the Wildcats, an |
See a YouTube video, uploaded on May 26, 2014,
of the Princeton Wildcats singing "Angel Eyes."
See also "Angel Eyes" and "Proginoskes" in this journal.
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Sunday, July 13, 2014
Useful Princeton
See…
- An Apple for Devlin (July 2)
- Thanks for having me on (July 11)
- Useful Idiot (Wikipedia article)
-
Useful Fictions (a Princeton mathematics
department Spring 2014 course)
"Numbers themselves are fictions, abstractions humans invented
to gain more control over the world." — Keith Devlin
Related material:
-
From today's New York Times Sunday Review ,
The World According to Maxwell Smart, Part 1
- Madonna, "Get Stupid," a political video from 2008
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Friday, January 10, 2014
Department of Corrections
The reference to David Justice at the beginning of
yesterday afternoon's post does not imply an
endorsement of all his writings. For instance, a
Justice post from yesterday contains the following—
Correction—
The above author name and page number are wrong.
Related to the above "fundamental theme" —
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Monday, June 3, 2013
New Yorker Art
New Yorker editor David Remnick at Princeton today
(from a copy of his prepared remarks):
"Finally, speaking of fabric design…."
I prefer Tom and Harold:
Tom Wolfe in The Painted Word —
"I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little 'fuliginous flatness' here … a little
'action painting' there … and some of that 'all great art
is about art' just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period…."
Harold Rosenberg in The New Yorker —
Tom's book seems to be repeating, in 1975, what Harold said better in 1969.
"Finally, speaking of fabric design…."
Note "fabric design" in Rosenberg's words on philistine views of the art of Noland.
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For Princeton’s Class Day
Triple Threat
"'Mr. Remnick's work is smart, funny and insightful —
a triple threat Class Day speaker!' said Class Day
co-chair Lily Alberts." — News at Princeton
Related material: David Remnick on Miss Gould.
See also Remnick and Miss Gould in this journal.
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Saturday, March 3, 2012
Big Art
For Women's History Month—
The Beam of Pink Light
From a post linked to on Lyxe's upload date, Feb. 6, 2012—
“… with primitives the beginnings of art, science, and religion
coalesce in the undifferentiated chaos of the magical mentality….”
— Carl G. Jung, “On the Relation of Analytical Psychology to Poetry,”
Collected Works, Vol. 15, The Spirit in Man, Art, and Literature,
Princeton University Press, 1966, excerpted in
Twentieth Century Theories of Art, edited by James M. Thompson.
See also the NY Lottery for St. Luke's Day, 2011, publication date
of the new edition of Philip K. Dick's VALIS quoted above.
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Sunday, November 17, 2024
Weyl, Symmetry, and the MOG
(HTML version of an earlier post)
Some historical background for a new book by Robert T. Curtis,
The Art of Working with the Mathieu Group M24 —
"Space is another example of an entity endowed with a structure.
Here the elements are points, and the structure is established
in terms of certain basic relations between points such as:
A, B, C lie on a straight line, AB is congruent CD, and the like.
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 its 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. After that you may start to investigate
symmetric configurations of elements, i.e. configurations which are
invariant under a certain subgroup of the group of all automorphisms;
and it may be advisable, before looking for such configurations,
to study the subgroups themselves, e.g. the subgroup of those
automorphisms which leave one element fixed, or leave two distinct
elements fixed, and investigate what discontinuous or finite subgroups
there exist, and so forth."
— Hermann Weyl, Symmetry, Princeton University Press, 1952.
(Page 144 in the Princeton Science Library edition of 1989.)
This square's automorphism group
has 322,560 transformations.
— The diamond theorem of Steven H. Cullinane.
This rectangle's automorphism group
has 244,823,040 transformations.
— The Miracle Octad Generator (MOG) of Robert T. Curtis.
The rectangle's automorphism group contains the
square's as a subgroup. The square's automorphism
group leaves invariant a set of 30 eight-subsquare sets
called affine hyperplanes. The rectangle's automorphism
group leaves invariant a set of 759 eight-subsquare sets
called octads.
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(HTML version of an earlier post)
(HTML version of an earlier post)
Friday, November 17, 2023
Classicism Continued: An Apotheosis of Modernity
|
From Chapter 23, "Poetry," by Adam Parkes, in Writing in 1910–11, the English poet and critic T. E. Hulme claimed that the two major traditions in poetry, romanticism and classicism, were as different as a well and a bucket. According to the romantic party, Hulme explained, humankind is “intrinsically good, spoilt by circumstance”; that is, our nature is “a well, a reservoir full of possibilities.” For the classical party, however, human nature is “like a bucket”; it is “intrinsically limited, but disciplined by order and tradition to something fairly decent” (Hulme 1987: 117). But it was not only that romanticism and classicism were as dissimilar as a well and a bucket; their contents were different, too. To draw water from the well of romanticism was, in effect, to pour a “pot of treacle over the dinner table,” while the classical bucket was more likely to be full of little stones – or jewels, perhaps. Romanticism, in Hulme’s view, was the result of displaced religious fervor; it represented the return of religious instincts that the “perverted rhetoric of Rationalism” had suppressed, so that “concepts that are right and proper in their own sphere are spread over, and so mess up, falsify and blur the clear outlines of human experience” (Hulme 1987: 118). Classicism, by contrast, traded in dry goods – dry, hard goods, to be precise. Hulme left little doubt as to which side he was on. “It is essential to prove,” he argued, “that beauty may be in small, dry things. The great aim is accurate, precise and definite description. . . . I prophesy that a period of dry, hard, classical verse is coming” (Hulme 1987: 131–3). If by “dry, hard, classical verse” Hulme meant poems looking like the fragments of Sappho, he didn’t have to wait long to see his prophecy fulfilled.
The hard sand breaks,
Far off over the leagues of it, 228
playing on the wide shore, So wrote Hilda Doolittle in “Hermes of the Ways,” the first poem that she signed “H. D., Imagiste” at the behest of her fellow American expatriate Ezra Pound. From Pound’s perspective, the Imagist movement that he co-founded in 1912 with H. D. and the English poet Richard Aldington was finished well before the First World War began in August 1914; throughout this war-torn decade, however, Imagism continued to spawn the poetry of “small, dry things” whose coming Hulme had predicted a few years before. Indeed, modernist poets weren’t content merely to break down the extended heroic narratives – the “spilt religion,” as Hulme put it – of their treacly nineteenthcentury predecessors; they insisted on breaking down small things into ever-smaller particles and subparticles. This logic of disintegration is clearly at work in poems like “Hermes of the Ways,” where each line is metrically unique, creating a sense of perpetual freshness – an apotheosis of modernity, as it were. REFERENCE Hulme, T. E. (1987). Speculations: Essays on Humanism and the Philosophy of Art, ed. Herbert Read. London and New York: Routledge and Kegan Paul. First published 1924. |
Compare and contrast:
Jeremy Gray,
Plato's Ghost: The Modernist Transformation of Mathematics,
Princeton University Press, first edition Sept. 22, 2008 —
"Here, modernism is defined as an autonomous body of ideas,
having little or no outward reference, placing considerable emphasis
on formal aspects of the work and maintaining a complicated—
indeed, anxious— rather than a naïve relationship with the
day-to-day world, which is the de facto view of a coherent group
of people, such as a professional or discipline-based group
that has a high sense of the seriousness and value of what it is
trying to achieve. This brisk definition…."
(Quoted at the webpage Solomon's Cube.)
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Saturday, June 10, 2023
Green, Orange, Black
The colors surrounding Watson's body in the above
"bandeau" photo suggest a review. A search in this journal
for Green+Orange+Black yields . . .
In the above image, the "hard core of objectivity" is represented
by the green-and-white eightfold cube. The orange and black are,
of course, the Princeton colors.
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Saturday, February 18, 2023
Probing the Wormhole
In memory of illustrator Istvan Banyai,
who reportedly died on Dec. 15, 2022 . . .
https://www.nytimes.com/2023/02/18/
arts/istvan-banyai-dead.html —
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Wednesday, January 11, 2023
How the Darkness Gets In
|
THE PHILOSOPHY OF RUDOLF CARNAP
EDITED BY PAUL ARTHUR SCHILPP
. . . . |
See also . . .
The extraordinary consequences of Einstein’s universe:
Relativity shatters our experience of time
9th January 2023
By Michael David Silberstein
"Professor of Philosophy at Elizabethtown College
and co-athor [sic] of Emergence in Context:
A treatise of twentry [sic] first-century natural philosophy
(Oxford University Press, 2022)."
"… the experience that there is something special about
the character of the present moment. This is what presumably
lead [sic] Einstein to say that
'there is something essential about the Now
which is just outside the realm of science.' "
Silberstein does not give any source for his quotation.
But see the passage from Carnap above.
I do not recommend taking Carnap's — or Silberstein's —
word for anything.
The source of Silberstein's remarks is a publication of an
organization called "Institute of Art and Ideas," or IAI.
Wikipedia on that organization:
"The IAI is responsible for organising the bi-annual festival
HowTheLightGetsIn, the biggest philosophy and music
festival in the world* aimed at 'tackling the dearth of philosophy
in daily life,' in addition to monthly IAI Live events."
* Maya Oppenheim (7 September 2021):
"HowTheLightGetsIn: The world's largest philosophy
and music festival to ask life's big questions."
The Independent.
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Thursday, November 24, 2022
The Drum Machine
"A struggling music producer sells his soul to a 1970s drum machine."
— Summary of a short film by Kevin Ignatius, "Hook Man."
The music producer pawns his current drum device
and acquires a demonic 1970s machine.
Artistic symbolism —
The 16-pad device at left may be viewed by enthusiasts of ekphrasis
as a Galois tesseract, and the machine at right as the voice of
Hal Foster, an art theorist who graduated from Princeton in 1977.
For an example of Foster's prose style, see
the current London Review of Books.
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Friday, November 4, 2022
Dark Fields Lyric
Heard Thursday evening in the dark fields of the republic —
"Rebel rebel, you've torn your dress
Rebel rebel, your face is a mess
Rebel rebel, how could they know?
Hot tramp, I love you so!"
Addendum — 10:30 AM ET on 5 Nov. 2022 —
Addendum — 5:45 PM ET on 4 Nov. 2022 —
The Triangle Club of Sadness
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Wednesday, June 22, 2022
Occupy Space Continues.
Alternate Title —
Types of Ambiguity:
The Circle in the Triangle,
the Singer in the Song.
From an excellent June 17 Wall Street Journal review of a new
Isaac Bashevis Singer book from Princeton University Press —
" 'Old Truths and New Clichés,' a collection of 19
prose articles, most appearing in English for the
first time, reveals that Singer was as consummate
an essayist as he was a teller of tales." — Benjamin Balint
From a search in this journal for Singer —
Related material —
From a post of June 2, "Self-Enclosing" —
|
"… the self-enclosing processes by which late 20th-century
— Colin Burrow in the June 9, 2022 issue |
From the December 14, 2021, post Notes on Lines —
The triangle, a percussion instrument that was
featured prominently in the Tom Stoppard play
"Every Good Boy Deserves Favour."
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Friday, May 6, 2022
The Hat Tip
Related material —
"A good, involving mystery featuring strong characters and
prose as smooth as the brim of a fedora, this novel makes
smart points about writing, publishing and the cult of mysteries."
— Review of A Smile on the Face of the Tiger
See also . . .
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Sunday, April 17, 2022
Annals of Iconic Simplicity
The New York Times today has an obituary for
Kevin Lippert, the founder and publisher of
Princeton Architectural Press, who reportedly
died at 63 on March 29, 2022.
“'There was a space between the academic,
theory-heavy M.I.T. Press and the coffeetableism
of Rizzoli,' Mr. Lamster wrote, adding that
Princeton Architectural Press would fill the gap
with 'the voice of the young practitioner.'
Mr. Lippert championed emerging architects.
He published Steven Holl’s seminal architectural
manifesto, 'Anchoring,' in 1989, and wrote the
introduction to the book of the same name.
Mr. Holl, in a tribute to Mr. Lippert on his website,
called him 'a committed intellectual and impresario
for the culture of architecture.'”
— Katharine Q. Seelye, April 17, 2022, 2:21 p.m. ET
From the cited tribute to Lippert on Holl's website —
"An excerpt from his publisher’s foreword to Anchoring :
In its iconic simplicity, his work seems to be about
the language of architecture, not in the allusive sense
used by postmodernists nor in the paradigmatic sense
used by so-called 'deconstructivists' but at the level of
essences of tropes and morphs… He is the only
American architect of his generation to be directly
influenced by the main lines in modern philosophy and
music, that is to say, by the line leading from Husserl
through to Heidegger and by separate achievements
of Bartok and Schonberg ."
Actually, although the above "iconic simplicity" passage,
up to the ellipsis after "morphs," is from the foreword
by Lippert, the references that follow the ellipsis — to
Husserl, Heidegger, Bartok, and Schonberg — are not
from Lippert's foreword, but from the introduction by
one Kenneth Frampton —
From Google Books:
Bibliographic data —
Another architectural memorial, from the reported date of Lippert's death —
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Tuesday, August 10, 2021
Ex Fano Apollinis
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
.jpg)
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
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Tuesday, December 22, 2020
Small Venues
“… her art was rarely exhibited until the 1970s,
and then only sporadically and in small venues . . . .”
— New York Times obituary suggested by
today’s review,
https://www.nytimes.com/2020/12/22/
arts/artists-who-died-2020.html
“No ordinary venue.” — Song lyric
Related material now linked to in the previous post —
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Thursday, June 4, 2020
Times Literary Supplement
See also mentions of Justin E. H. Smith in this journal, including . . .
Monday, June 4, 2012
Rigor and Respect
“… Western academic philosophy will likely come to appear
utterly parochial in the coming years if it does not find a way
to approach non-Western traditions that is much more rigorous
and respectful than the tokenism that reigns at present.”
— Justin E. H. Smith in the New York Times philosophy
column “The Stone” yesterday
For example—
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Wednesday, August 28, 2019
Tiger’s Leap to 1905
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Monday, May 6, 2019
In Memoriam Goro Shimura (d. May 3, 2019)
From Richard Taylor, "Modular arithmetic: driven by inherent beauty
and human curiosity," The Letter of the Institute for Advanced Study [IAS],
Summer 2012, pp. 6– 8 (links added) :
"Stunningly, in 1954, Martin Eichler (former IAS Member)
found a totally new reciprocity law . . . .
Within less than three years, Yutaka Taniyama and Goro Shimura
(former IAS Member) proposed a daring generalization of Eichler’s
reciprocity law to all cubic equations in two variables. A decade later,
André Weil (former IAS Professor) added precision to this conjecture,
and found strong heuristic evidence supporting the Shimura-Taniyama
reciprocity law. This conjecture completely changed the development of
number theory."
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Saturday, March 16, 2019
Multifaceted Narrative
"Here, modernism is defined as an autonomous body
of ideas, having little or no outward reference, placing
considerable emphasis on formal aspects of the work
and maintaining a complicated—indeed, anxious—
rather than a naïve relationship with the day-to-day
world, which is the de facto view of a coherent group
of people, such as a professional or discipline-based
group that has a high sense of the seriousness and
value of what it is trying to achieve. This brisk definition…."
— Jeremy Gray, Plato's Ghost: The Modernist
Transformation of Mathematics , Princeton, 2008
"Even as the dominant modernist narrative was being written,
there were art historians who recognized that it was inaccurate.
The narrative was too focused on France . . . . Nor was it
correct to build the narrative so exclusively around formalism;
modernism was far messier, far more multifaceted than that."
— Jane Kallir, https://www.tabletmag.com/
jewish-arts-and-culture/visual-art-and-design/
269564/the-end-of-middle-class-art,
quoted here on the above date — Sept. 11, 2018.
From some related Log24 posts —
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Wednesday, March 6, 2019
The Relativity Problem and Burkard Polster
|
From some 1949 remarks of Weyl— "The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time." — Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541 Weyl in 1946—: "This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them." — Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16 |
For some context, see Relativity Problem in this journal.
In the case of PG(3,2), there is a choice of geometric models
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.
The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.
For comparison of the two models, see a figure posted here on
May 21, 2014 —
Labeling the Tetrahedral Model (Click to enlarge) —
"Citation needed" —
The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)," a "reliable source."
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Tuesday, June 19, 2018
Death on Father’s Day
From the University of Notre Dame in an obituary dated June 17 —
|
Timothy O’Meara, provost emeritus, Kenna Professor of Mathematics Emeritus and Trustee Emeritus at the University of Notre Dame, died June 17. He was 90.
A member of the Notre Dame faculty since 1962, O’Meara twice served as chairman of the University’s mathematics department and served as its first lay provost from 1978 to 1996.
He was graduated from the University of Cape Town in 1947 and earned a master’s degree in mathematics there the following year. Earning his doctoral degree from Princeton University in 1953, he taught at the University of Otago in New Zealand from 1954 to 1956 before returning to Princeton where he served on the mathematics faculty and as a member of the Institute for Advanced Study for the next six years.
In addition to his mathematical teaching and scholarship, he published magisterial works, including “Introduction to Quadratic Forms,” “Lectures on Linear Groups,” “Symplectic Groups” and “The Classical Groups and K-Theory,” co-authored with Alexander J. Hahn, professor of mathematics emeritus at Notre Dame and a former O’Meara doctoral student. |
Related material (update of 9:20 PM ET on June 19) —
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Tuesday, April 24, 2018
Alma Maman
"Almost 9 meters tall, Maman is one of the most ambitious
of a series of sculptures by Bourgeois that take as their subject
the spider, a motif that first appeared in several of the artist's
drawings in the 1940s and came to assume a central place in
her work during the 1990s. Intended as a tribute to her mother,
who was a weaver, Bourgeois's spiders are highly contradictory
as emblems of maternity: they suggest both protector and predator—
the silk of a spider is used both to construct cocoons and to bind prey—
and embody both strength and fragility."
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Saturday, February 17, 2018
Frosties: A Sequel to “Frozen”
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Friday, February 16, 2018
Two Kinds of Symmetry
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.
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Saturday, February 10, 2018
Into the Upside Down
(Title suggested by the TV series Stranger Things )
" 'Untitled' (2016) is the most recent painting in the show
and includes one of Mr. Johns’s recurring images of a ruler."
— Image caption in an article by Deborah Solomon
in The New York Times online, Feb. 7, 2018
From a Log24 search for "Ruler" —
Related art —
See also, in this journal, Magic Mountain and Davos.
Einstein and Thomas Mann, Princeton, 1938
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Friday, November 10, 2017
Annals of Rarefied Scholarship
From Cambridge Core, suggested by a reference to
that website in the previous post and by the following
bibliographic data . . .
https://doi.org/10.1017/fmp.2016.5
Downloaded from https://www.cambridge.org/core
on 10 Nov 2017 at 19:06:19
See Conwell + Princeton in this journal.
Related art —
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Tuesday, October 3, 2017
Show Us Your Wall
From Monday morning's post Advanced Study —
"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."
— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8
Related news from today's online New York Times —
A heading from the above screenshot: "SHOW US YOUR WALL."
This suggests a review of a concept from Galois geometry —
(On the wall — a Galois-geometry inscape .)
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Tuesday, September 5, 2017
Florence 2001
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
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Thursday, August 31, 2017
A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
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Thursday, May 18, 2017
Bit Plot
From a May 15 review of a new book by Douglas Coupland, author of
the 1991 book Generation X: Tales for an Accelerated Culture .
"Minimalists are actually extreme hoarders:
they hoard space." — Douglas Coupland
The title of Coupland's new book suggests a review of Schmeikal
in this journal …
Coupland's above remark on hoarders suggests a look at
a wealthy California collector whom, were he not wealthy,
some might call a hoarder.
“I buy things because they strike an emotional bell,
they appeal to my curiosity, to the thrill of discovery
of the extraordinary in the ordinary,” Mr. Cotsen told
The Denver Post in 1998. “They appeal to my sense
of humor, and to my search for the beauty in simplicity.”
He added, “I decided I had a collection when there was
no more space to put anything.”
By the time he died at 88 on May 8 in Beverly Hills, Calif.,
Mr. Cotsen (pronounced COAT-zen) had donated about
half of the material in his collections to institutions like the
Asian Art Museum in San Francisco, Princeton University
and the Museum of International Folk Art in Santa Fe, N.M.
— Richard Sandomir in the online New York Times , May 17
Cotsen reportedly died at 88 on May 8.
See also this journal on that date —
|
Monday, May 8, 2017
New Pinterest Board
|
Comments Off on Bit Plot
Sunday, March 26, 2017
Four-Year* Date
"Eigenvalues. Fixed points. Stable equilibria.
Mathematicians like things that stay put.
And if they can't stay put, the objects of study
should at least repeat themselves on a regular basis. . . ."
— Barry Cipra, "A Moveable Feast," SIAM News , Jan. 14, 2006
Tuesday, March 18, 2014
|
* For a full four years, see also March 18, 2013.
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Monday, March 6, 2017
Bullshit Studies
From The Chronicle of Higher Education on March 2, 2017 —
|
These days, in a world totally dependent on microprocessors, lasers, and nanotechnology, it has been estimated that 30 percent of the U.S. gross national product is based on inventions made possible by quantum mechanics. With the booming high-tech industry and the expected advent of quantum computers, this percentage will only grow. Within a hundred years, an esoteric theory of young physicists became a mainstay of the modern economy. It took nearly as long for Einstein’s own theory of relativity, first published in 1905, to be used in everyday life in an entirely unexpected way. The accuracy of the global positioning system, the space-based navigation system that provides location and time information in today’s mobile society, depends on reading time signals of orbiting satellites. The presence of Earth’s gravitational field and the movement of these satellites cause clocks to speed up and slow down, shifting them by 38 milliseconds a day. In one day, without Einstein’s theory, our GPS tracking devices would be inaccurate by about seven miles. — Robbert Dijkgraaf, Director, Institute for Advanced Study, Princeton |
The above paragraphs are clearly propaganda, not physics.
For "It has been estimated," see …
The "without Einstein 's theory" statement may or may not be correct.
See the lengthy discussion at …
http://physics.stackexchange.com/questions/1061/
why-does-gps-depend-on-relativity .
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Sunday, February 12, 2017
Colorful Tales
“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”
— Hermann Weyl, Philosophy of Mathematics and
Natural Science , Princeton, 1949, p. 237
Harvard University Press on the late Angus Fletcher, author of
The Topological Imagination and Colors of the Mind —
From the Harvard webpage for Colors of the Mind —
Angus Fletcher is one of our finest theorists of the arts,
the heir to I. A. Richards, Erich Auerbach, Northrop Frye.
This… book… aims to open another field of study:
how thought— the act, the experience of thinking—
is represented in literature.
. . . .
Fletcher’s resources are large, and his step is sure.
The reader samples his piercing vision of Milton’s
Satan, the original Thinker,
leaving the pain of thinking
as his legacy for mankind.
A 1992 review by Vinay Dharwadker of Colors of the Mind —
See also the above word "dianoia" in The Echo in Plato's Cave.
Some context …
This post was suggested by a memorial piece today in
the Los Angeles Review of Books —
A Florilegium for Angus Fletcher
By Kenneth Gross, Lindsay Waters, V. N. Alexander,
Paul Auster, Harold Bloom, Stanley Fish, K. J. Knoespel,
Mitchell Meltzer, Victoria Nelson, Joan Richardson,
Dorian Sagan, Susan Stewart, Eric Wilson, Michael Wood
Fletcher reportedly died on November 28, 2016.
"I learned from Fletcher how to apprehend
the daemonic element in poetic imagination."
— Harold Bloom in today's Los Angeles florilegium
For more on Bloom and the daemonic, see a Log24 post,
"Interpenetration," from the date of Fletcher's death.
Some backstory: Dharwadker in this journal.
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Thursday, December 22, 2016
Constructivist Witness
The title refers to a philosophy of mathematics.
For those who prefer metaphor… Folk Etymology.
See also Stages of Math at Princeton's
Institute for Advanced Study in March 2013 —
— and in this journal starting in August 2014.
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Wednesday, December 14, 2016
Graveyard Roses
Two deaths on Tuesday, Dec. 13, 2016 —
In memory of game show figure Alan Thicke —
In memory of game theory author Thomas Schelling —
Barbara Rose in a Log24 search for Princeton + Art.
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Tuesday, November 15, 2016
A Paris Review
The half-hour referred to here was from 12 PM ET
to 12:30 PM ET on Friday, April 4, 2014 …
12 PM at Log24 —
12:30 PM at Princeton —
The New York Times on an art lecturer who died on Nov. 9 —
She became a Vogue correspondent in postwar Paris
and worked for art magazines before starting her own,
the celebrated L’Oeil (The Eye).
See also Obituary Metaphysics from November 11th —
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Tuesday, October 25, 2016
Tuesday, August 23, 2016
Puritan Contemplation:
For an authority on Japanese art —
Text Tiles*
Compare to and contrast with …
Remarks on art, contemplation. and Puritanism
from a recent Princeton University Press book —
“Lucy Lippard distinguished Asian art
(ego-less and contemplative)
from New York Minimalism
(moralistic and puritanical).”
— Mathematics and Art ,
Princeton U. Press, Fall 2015
* Update of Aug. 24, 2016 — See also Nov. 2, 2014.
Comments Off on Puritan Contemplation:
Friday, August 12, 2016
Dustbucket Physics
Peter Galison, a Harvard professor, is a defender of
the Vienna Circle and the religion of Scientism.
From Galison's “Structure of Crystal, Bucket of Dust,” in
Circles Disturbed: The Interplay of Mathematics and Narrative ,
edited by Apostolos Doxiadis and Barry Mazur, pp. 52-78
(Princeton: Princeton U. Press, 2012) …
Galison's final paragraph —
"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginning-end of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.' "
For fans of Scientism who prefer more colorful narratives —
Comments Off on Dustbucket Physics
Friday, June 24, 2016
Contrast
From a work cited in the previous post —
"… representation of hell and the horrors
of the burial ground are missing."
— Page 384 of Joseph Campbell's The Mythic Image ,
Princeton University Press, 1981
(First published in 1974)
For those who regret the above omission …
A review of a book published in 1977 —
"Its materials are fear and death, hallucination
and the burning of souls."
The book's author reportedly died Thursday, June 23, 2016.
See also, from 11 AM ET that day, "Raiders of the Lost Code."
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Tuesday, April 12, 2016
The Remnick Remark
A remark by New Yorker editor David Remnick
at Princeton on June 3, 2013 —
"Finally, speaking of fabric design. . . ."
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Monday, April 11, 2016
Gospel of the Nobodies
"Principles before personalities" — AA saying
Principles
From an April 8 Princeton obituary of a mathematician —
" Moore embodied a 'Princeton style' that made him
a challenging and influential presence in the careers
of his students, said Joseph Neisendorfer, a professor
of mathematics at the University of Rochester who
received his Ph.D. in mathematics from Princeton in
1972. Because of Moore's style, his students would
write theses that 'almost without exception' were
significant advances in mathematics, Neisendorfer said.
'There's a certain Princeton style that focuses on
precision, centrality and simplicity. He was a superb
mathematician and he exercised a lot of influence
by imparting his style to his students,' Neisendorfer said.
'He epitomized the Princeton style.' "
Personalities
Gospel of the Nobodies
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Tuesday, April 5, 2016
Spring Play
The spring play this March at Princeton's McCarter Theatre Center
was Agatha Christie's "The Mousetrap."
In related news —
See as well, in this journal, a post from the date pictured above,
that of the Disneyland Diamond Celebration on May 22, 2015 —
Comments Off on Spring Play
Wednesday, March 30, 2016
Romanesque
From a New York Times obituary
of Ellsworth Kelly by Holland Cotter —
"The anonymous role of
the Romanesque church artist
remained a model."
See as well
Note the contradiction between the URL date (last Monday's)
and the printed date below it (that of Epiphany 2016).
Who's trolling whom?
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Monday, February 1, 2016
The Hiroshima Preprint
This morning at 11:44 I happened upon …
This was published as …
Toshiyuki Katsura, Shigeyuki Kondo, Ichiro Shimada,
"On the supersingular K3 surface in characteristic 5 with Artin invariant 1,"
Michigan Mathematical Journal , vol. 63, issue 4 (Dec. 2014), 803–844.
Related material from later today —
See also earlier Log24 remarks on the Hoffman-Singleton graph
and a remark on geometry for Princeton.
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Saturday, November 21, 2015
Brightness at Noon*
A recent not-too-bright book from Princeton —
Some older, brighter books from Tony Zee —
Fearful Symmetry (1986) and
Quantum Field Theory in a Nutshell (2003).
* Continued.
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Thursday, November 5, 2015
The Monster
In memory of Princeton mathematician John Nash
"For the past six years all over the world
experts in the branch of abstract algebra
called group theory have been struggling
to capture a group known as the monster."
—Martin Gardner, Scientific American , June 1980
"When the Hawkline Monster moved to get a better view
of what was happening, the shadow, after having checked
all the possibilities of light, had discovered a way that it
could shift itself in front of the monster, so that the monster
at this crucial time would be blinded by darkness for a few
seconds, did so, causing confusion to befall the monster.
This was all that the shadow could do and it hoped that this
would give Greer and Cameron the edge they would need
to destroy the Hawkline Monster using whatever plan they
had come up with, for it seemed that they must have a plan
if they were to have any chance at all with the monster and
they did not seem like fools.
When Cameron yelled at Greer, the shadow interpreted this
as the time to move and did so. It obscured the vision of the
Hawkline Monster for a few seconds, knowing full well that if
the monster were destroyed it would be destroyed, too, but
death was better than going on living like this, being a part of
this evil."
— Richard Brautigan, The Hawkline Monster , 1974
From the post For Scientific Witch Hunters of October 30,
an illustration from The Boston Globe —
From the post Colorful Story (All Souls' Day),
an Illustration from Google Book Search —
Earlier in Brautigan's tale …
" Everybody started to leave the parlor to go downstairs
and pour out the Hawkline Monster but just as
they reached the door and one of the Hawkline women
had her hand on the knob, Cameron said, 'Hold it for a
second. I want to get myself a little whiskey.' "
Comments Off on The Monster
Wednesday, November 4, 2015
Saturday, October 24, 2015
Sunday, August 16, 2015
Sunday School
The title of the previous post, "For Quantum Mystics,"
suggests a search in this journal for Quantum + Mystic.
That search in turn suggests, in particular, a review of
a post of October 16, 2007 — a discussion of the
P.T. Barnum-like phrase "deep beauty" used to describe
a topic under discussion at Princeton by physicists.
Princeton, by the way, serves to illustrate the "gutter"
mentioned by Sir Laurence Olivier in a memorable
classroom scene from 1962.
Comments Off on Sunday School
Tuesday, June 2, 2015
Thursday, April 23, 2015
Colorful Tale
(A sequel to yesterday's ART WARS and this
morning's De Colores )
“Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind– as Eddington puts it– the colorful tale
of the subjective storyteller mind.” — Hermann Weyl
(Philosophy of Mathematics and Natural Science ,
Princeton, 1949, p. 237)
See also Deathly Hallows.
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Saturday, April 11, 2015
The Starbird Manifesto
"But what was supposed to be the source of a compound's
authority? Why, the same as that of all new religious movements:
direct access to the godhead, which in this case was Creativity."
— Tom Wolfe, From Bauhaus to Our House
"Creativity is not a matter of magical inspiration."
— Burger and Starbird, The 5 Elements of Effective Thinking (2012)
Video published on Oct 19, 2012
"In this fifth of five videos, mathematics professor
Michael Starbird talks about the fifth element
in his new book, The 5 Elements of Effective Thinking ,
co-authored with Williams College professor
Edward B. Burger."
For more on the Starbird manifesto, see Princeton University Press.
An excerpt —
See also a post for Abel's Birthday, 2011 —
Midnight in Oslo — and a four-elements image from
the Jan. 26, 2010, post Symbology —
.
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Tuesday, February 17, 2015
Style News
Comments Off on Style News
Thursday, February 12, 2015
Dead Reckoning
Continued from yesterday.
The passage on Claude Chevalley quoted here
yesterday in the post Dead Reckoning was, it turns out,
also quoted by Peter Galison in his essay "Structure of Crystal,
Bucket of Dust" in Circles Disturbed: The Interplay of
Mathematics and Narrative (Princeton University Press, 2012,
ed. by Apostolos Doxiadis and Barry Mazur).
Galison gives a reference to his source:
"From 'Claude Chevalley Described by His Daughter (1988),'
in Michèle Chouchan, Nicolas Bourbaki: Faits et légendes
(Paris: Éditions du Choix, 1995), 36–40, translated and cited
in Marjorie Senechal, 'The Continuing Silence of Bourbaki:
An Interview with Pierre Cartier, June 18, 1997,'
Mathematical Intelligencer 1 (1998): 22–28."
Galison's essay compares Chevalley with the physicist
John Archibald Wheeler. His final paragraph —
"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginning-end of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.'"
See also posts tagged Wheeler.
Comments Off on Dead Reckoning
Monday, February 9, 2015
Overarching Symmetry
Continued from earlier posts.
|
The Washington Post online yesterday: "Val Logsdon Fitch, the Nebraska rancher’s son who shared the Nobel Prize for detecting a breakdown in the overarching symmetry of physical laws, thus helping explain how the universe evolved after the Big Bang, died Feb. 5 in Princeton, N.J. He was 91. His death was confirmed by Princeton University, where he had been a longtime faculty member and led the physics department for several years. Dr. Fitch and his Princeton colleague James Cronin received the Nobel Prize in physics in 1980 for high-energy experiments conducted in 1964 that overturned fundamental assumptions about symmetries and invariances that are characteristic of the laws of physics." — By Martin Weil |
Fans of synchronicity may prefer some rather
ig -Nobel remarks quoted here on the date
of Fitch's death:
"The Harvard College Events Board presents
Harvard Thinks Big VI, a night of big ideas
and thinking beyond traditional boundaries.
On Thursday February 5th at 8 pm in
Sanders Theatre …."
— Log24 post The Big Spielraum
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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 .
Comments Off on The Relaxed Field*
Thursday, January 29, 2015
Hard Problem
The Yale Daily News on Sept. 9, 2014 —
Related material on "the hard problem" of consciousness—
Wikipedia on the problem, and Tom Stoppard's first new
play in nine years, "The Hard Problem."
See also, in this journal, the posts of Sept. 9, 2014,
the date of the above Yale Daily News story
"Research Suggests New Consciousness Hub."
The above scene from the new Stoppard play
suggests also a review of Kulturkampf for Princeton.
Comments Off on Hard Problem
Monday, December 15, 2014
Mythic Metaphysics
Today’s 8:01 PM post quoted Husserl on
the perception of the cube.
Another approach to perception of the cube,
from Narrative Metaphysics on St. Lucia’s Day —
 See also Symplectic Structure and Stevens’s Rock. |
From today’s 11:29 AM post —
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
For some illuminating remarks on a mythical approach
to perception of the cube, see Gareth Knight on Schicksalstag 2012.
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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”> |
Comments Off on May We?
Saturday, December 13, 2014
Grim Pen
The title refers to remarks linked to this afternoon :
"An ingenious story line serves to convey
the mysteriousness of destiny, the 'grim pen'
of fate that encloses Hazel Shade."
— John Burt Foster Jr. on page 224 of
Nabokov's Art of Memory and European Modernism
(Princeton University Press, 1993)
Vide a relevant page on Wallace Stevens.
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Tuesday, December 2, 2014
Colorful Tale
"Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind— as Eddington puts it— the colorful tale
of the subjective storyteller mind."
— Hermann Weyl in Philosophy of Mathematics
and Natural Science , Princeton, 1949, p. 237
Tom Wolfe on art theorists in The Painted Word (1975) :
"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde understood, they
were not only theories but … hot news,
straight from the studios, from the scene."
The Weyl quote is a continuing theme in this journal.
The Wolfe quote appeared here on Nov. 18, 2014,
the reported date of death of Yale graduate student
Natasha Chichilnisky-Heal.
Directions to her burial (see yesterday evening) include
a mention of "Paul Robson Street" (actually Paul
Robeson Place) near "the historic Princeton Cemetery."
This, together with the remarks by Tom Wolfe posted
here on the reported day of her death, suggests a search
for "red green black" —
The late Chichilnisky-Heal was a student of political economy.
The search colors may be interpreted, if one likes, as referring
to politics (red), economics (green), and Robeson (black).
See also Robeson in this journal.
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Saturday, November 29, 2014
Mountain, Fountain
For the late Vladimir Nabokov, author of Pale Fire :
He took his article from a steel file:
"It's accurate. I have not changed her style.
There's one misprint–not that it matters much:
Mountain, not fountain. The majestic touch."
Click for a related Hollywood Reporter story.
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Wednesday, November 26, 2014
A Tetrahedral Fano-Plane Model
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by
Burkard Polster (Springer, 1998).
"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"
—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya
For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."
The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay"
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.
Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.
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Monday, November 3, 2014
The Rhetoric of Abstract Concepts
From a post of June 3, 2013:
New Yorker editor David Remnick at Princeton today
(from a copy of his prepared remarks):
“Finally, speaking of fabric design….”
I prefer Tom and Harold:
Tom Wolfe in The Painted Word —
“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”
Harold Rosenberg in The New Yorker (click to enlarge)—
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
— Steven H. Cullinane,
diamond theorem illustration
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Monday, September 29, 2014
Michaelmas Texts
This morning’s previous post quoted a sort of
invitation to damnation
from Princeton University Press:
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Michaelmas Mystery
Some related material in this journal: “Peter J. Cameron” + Magic.
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Friday, July 25, 2014
Magic for Jews
(Continued from April 8, 2013.)
See Two Blocks Short of a Design (May 5, 2011).
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Sunday, July 6, 2014
Game News
An essay linked to here on the date of Kuhn’s
death discussed the film “Good Will Hunting”:
“You can be sure that when an experienced movie director
like Gus Van Sant selects an establishing shot for the lead
character, he does so with considerable care, on the advice
of an expert.”
Establishing shots —
1. From a post of January 29, 2014:
2. From a post of April 12, 2011:
Parting shot —
From another post of January 29, 2014:
Note Watson‘s title advice.
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Friday, April 4, 2014
April in Paris
The Surreal Meets the Real
“Place Vendôme” (1998)
Odilon Redon, L’Oeil
FILM REVIEW
For Deneuve, a Setting In Which She Sparkles
By DAVE KEHR
Published: August 18, 2000 in The New York Times
“Named after the Parisian square that is home
to the Ritz Hotel, several haute couture boutiques
and some of the world’s most expensive jewelry shops,
‘Place Vendôme’ provides a perfect setting for
Catherine Deneuve, herself one of the French cinema’s
most precious gems.”
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Friday, March 14, 2014
The Search for Charles Wallace
The search in the previous post for the source of a quotation from Poincaré yielded, as a serendipitous benefit, information on an interesting psychoanalyst named Wilfred Bion (see the Poincaré quotation at a webpage on Bion). This in turn suggested a search for the source of the name of author Madeleine L'Engle's son Bion, who may have partly inspired L'Engle's fictional character Charles Wallace. Cynthia Zarin wrote about Bion in The New Yorker of April 12, 2004 that
"According to the family, he is the person for whom L’Engle’s insistence on blurring fiction and reality had the most disastrous consequences."
Also from that article, material related to the name Bion and to what this journal has called "the Crosswicks Curse"*—
"Madeleine L’Engle Camp was born in 1918 in New York City, the only child of Madeleine Hall Barnett, of Jacksonville, Florida, and Charles Wadsworth Camp, a Princeton man and First World War veteran, whose family had a big country place in New Jersey, called Crosswicks. In Jacksonville society, the Barnett family was legendary: Madeleine’s grandfather, Bion Barnett, the chairman of the board of Jacksonville’s Barnett Bank, had run off with a woman to the South of France, leaving behind a note on the mantel. Her grandmother, Caroline Hallows L’Engle, never recovered from the blow. ….
… The summer after Hugh and Madeleine were married, they bought a dilapidated farmhouse in Goshen, in northwest Connecticut. Josephine, born in 1947, was three years old when they moved permanently to the house, which they called Crosswicks. Bion was born just over a year later."
* "There is such a thing as a tesseract."
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Monday, March 10, 2014
Strategic Hamlets
From an obituary for a Kennedy advisor
who reportedly died at 94 on February 23, 2014*—
“He favored withdrawing rural civilians
into what he called ‘strategic hamlets’
and spraying defoliants to cut off
the enemy’s food supply.”
Other rhetoric: Hamlet and Infinite Space in this journal,
as well as King of Infinite Space , Part I and Part II.
These “King” links, to remarks on Coxeter and Saniga ,
are about two human beings to whom Hamlet’s
phrase “king of infinite space” has been applied.
The phrase would, of course, be more accurately
applied to God.
* The date of the ‘God’s Architecture’ sermon
at Princeton discussed in this afternoon’s post.
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God’s Architecture
Part I:
The sermon, “God’s Architecture,” at Nassau Presbyterian
Church in Princeton on Sunday, Feb. 23, 2014. (This is the
“sermon” link in last Sunday’s 11 AM ET Log24 post.)
An excerpt:
“I wonder what God sees when God looks at our church.
Bear with me here because I’d like to do a little architectural
redesign. I look up at our sanctuary ceiling and I see buttons.
In those large round lights, I see buttons. I wonder what would
happen if we unbutton the ceiling, Then I wonder if we were to
unzip the ceiling, pull back the rooftop, and God were to look in
from above – What does God see? What pattern, what design,
what shape takes place?” — Rev. Lauren J. McFeaters
Related material — All About Eve:
A. The Adam and Eve sketch from the March 8 “Saturday Night Live”
B. “Katniss, get away from that tree!” —
C. Deconstructing God in last evening’s online New York Times .
Part II:
“Heavensbee!” in the above video, as well as Cartier’s Groundhog Day
and Say It With Flowers.
Part III:
Humans’ architecture, as described (for instance) by architecture
theorist Anne Tyng, who reportedly died at 91 on Dec. 27, 2011.
See as well Past Tense and a post from the date of Tyng’s death.
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Wednesday, January 22, 2014
A Riddle for Davos
Einstein and Thomas Mann, Princeton, 1938
See also the life of Diogenes Allen, a professor at Princeton
Theological Seminary, a life that reportedly ended on the date—
January 13, 2013— of the above Log24 post.
January 13 was also the dies natalis of St. James Joyce.
Some related reflections —
"Praeterit figura huius mundi " — I Corinthians 7:31 —
|
Conclusion of of "The Dead," by James Joyce— The air of the room chilled his shoulders. He stretched himself cautiously along under the sheets and lay down beside his wife. One by one, they were all becoming shades. Better pass boldly into that other world, in the full glory of some passion, than fade and wither dismally with age. He thought of how she who lay beside him had locked in her heart for so many years that image of her lover's eyes when he had told her that he did not wish to live. Generous tears filled Gabriel's eyes. He had never felt like that himself towards any woman, but he knew that such a feeling must be love. The tears gathered more thickly in his eyes and in the partial darkness he imagined he saw the form of a young man standing under a dripping tree. Other forms were near. His soul had approached that region where dwell the vast hosts of the dead. He was conscious of, but could not apprehend, their wayward and flickering existence. His own identity was fading out into a grey impalpable world: the solid world itself, which these dead had one time reared and lived in, was dissolving and dwindling. A few light taps upon the pane made him turn to the window. It had begun to snow again. He watched sleepily the flakes, silver and dark, falling obliquely against the lamplight. The time had come for him to set out on his journey westward. Yes, the newspapers were right: snow was general all over Ireland. It was falling on every part of the dark central plain, on the treeless hills, falling softly upon the Bog of Allen and, farther westward, softly falling into the dark mutinous Shannon waves. It was falling, too, upon every part of the lonely churchyard on the hill where Michael Furey lay buried. It lay thickly drifted on the crooked crosses and headstones, on the spears of the little gate, on the barren thorns. His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead. |
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Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
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Tuesday, December 24, 2013
Through a Mirror, Darkly
Review of a book first published in 1989—
Reality's Mirror: Exploring the Mathematics of Symmetry —
"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and up-to-date. The author is to be congratulated on
a job well done." — Martin Gardner
A completely different person whose name
mirrors that of the Mathematics of Symmetry author —
See also this journal on the date mentioned in the Princetonian .
"Always with a little humor." — Yen Lo
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Sunday, November 24, 2013
Wednesday, August 14, 2013
The Gift
"Give 'em hell." — Ben Bernanke at Princeton's Baccalaureate, 2013
Some background — Janet Leigh and the Museum of Modern Art
"The Varnedoe Debacle," by Hilton Kramer (Dec. 1991)
Hell… Hell. — Sinatra in The Manchuran Candidate
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Thursday, June 13, 2013
Gate
"Eight is a Gate." — Mnemonic rhyme
Today's previous post, Window, showed a version
of the Chinese character for "field"—
This suggests a related image—
The related image in turn suggests…
Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.
Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.
As was the I Ching. A related structure:
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Sunday, April 21, 2013
Stoppard Update
Meanwhile…
Daily Princetonian @princetonian
Related material:
A sermon by the man named today the new President of Princeton.
The sermon is from October 7, 2012. See also Log24 on that date.
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Monday, April 8, 2013
Magic for Jews
A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.
Here is such a look (click to enlarge):
I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)
The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not be encouraged to waste their time on this topic.
For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems."
Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .
Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.
Related material from today's Daily Princetonian :
See also Nash + Princeton in this journal.
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Saturday, March 2, 2013
Life of Pi
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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.
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Tuesday, January 1, 2013
The Simplest Situation
Thanks to a Harvard math major for the following V. I. Arnold quote
in a weblog post yesterday titled "Abstraction and Generality"—
"… the author has attempted to adhere to the principle of
minimal generality, according to which every idea should first
be clearly understood in the simplest situation;*
only then can the method developed be extended to
more complicated cases.
— Vladimir I. Arnold, Lectures on Partial Differential Equations
(Russian edition 1997; English translation 2004),
Preface to the second Russian edition
Thanks also to the math major for his closing post today.
* For instance… Natalie Angier's New Year's meditation
on a Buddha Field—
"… the multiverse as envisioned in Tibetan Buddhism,
'a vast system of 1059 [sic ; corrected to 10^59 on Jan. 3]
universes, that together are called a Buddha Field,' said
Jonathan C. Gold, who studies Buddhist philosophy at
Princeton."
— versus a search in this journal for "Japanese character" that yields…
Japanese character
for "field"
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Thursday, November 22, 2012
Finite Relativity
(Continued from 1986)
|
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
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Saturday, June 16, 2012
Chiral Problem
In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—
Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—
"ANDOVER–HARVARD THEOLOGICAL LIBRARY"
Detail from Knowles's obituary in this morning's New York Times—
William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.
Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
From Pilate Goes to Kindergarten—
The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—
Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?
A paper written in a somewhat similar spirit—
"Chiral Tetrahedrons as Unitary Quaternions"—
ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….
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Sunday, May 20, 2012
Monday, April 9, 2012
Eightfold Cube Revisited
A search today (Élie Cartan's birthday) for material related to triality*
yielded references to something that has been called a Bhargava cube .
Two pages from a 2006 paper by Bhargava—
Bhargava's reference [4] above for "the story of the cube" is to…
Higher Composition Laws I:
A New View on Gauss Composition,
and Quadratic Generalizations
Manjul Bhargava
The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217-250
Published by: Annals of Mathematics
Article Stable URL: http://www.jstor.org/stable/3597249
A brief account in the context of embedding problems (click to enlarge)—
For more ways of slicing a cube,
see The Eightfold Cube —
* Note (1) some remarks by Tony Smith
related to the above Dynkin diagram
and (2) another colorful variation on the diagram.
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Friday, January 27, 2012
Mathematics and Narrative (continued)
Princeton University Press on a book it will publish in March—
Circles Disturbed: The Interplay of Mathematics and Narrative
"Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier— 'Don't disturb my circles'— words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities."
Timeline of the Marvel Cinematic Universe — Norway, March 1942—
"The Red Skull finds the Tesseract, a cube of strange power,
said to be the jewel of Odin’s treasure room, in Tonsberg Norway.
(Captain America: The First Avenger)"
Tesseracts Disturbed — (Click to enlarge)
Detail of Tesseracts Disturbed —
Narrative of the detail—
See Tesseract in this journal and Norway, May 2010—
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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
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Tuesday, September 20, 2011
Relativity Problem Revisited
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls the relativity problem—
“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”
– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research“
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16
…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M 24. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….
Footnote of Sept. 20, 2011:
* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters….”
See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.
Some related articles by Curtis:
R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429
R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12 and M 24” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’ (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396
R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57
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Thursday, August 4, 2011
Midnight in Oslo
For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)
Plato's Diamond
The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—
Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate…. … Both in the Republic and in the Sophist there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno that the square on the diagonal is double the original square in area." – William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985 Plato's paradigm in the Meno—
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change—
Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |
Related material: Plato's Diamond by Oslo artist Josefine Lyche.
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"Remember me to her."
— Closing words of the Algis Budrys novel Rogue Moon .
Background— Some posts in this journal related to Abel or to random thoughts from his birthday.
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Monday, June 20, 2011
The Search for Invariants
The title of a recent contribution to a London art-related "Piracy Project" begins with the phrase "The Search for Invariants."
A search for that phrase elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."
Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."
Cassirer's concepts seem related to Weyl's famous remark that
“Objectivity means invariance with respect to the group of automorphisms.”
—Symmetry (Princeton University Press, 1952, page 132)
See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.
Update of 6 to 7:50 PM June 20, 2011—
Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.
Another source on Cassirer, invariance, and objectivity—
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
A search in Weyl's Symmetry for any reference to Ernst Cassirer yields no results.
* Published in French in 1938.
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Friday, March 4, 2011
Ageometretos Medeis Eisito*
Your mission, should you choose to accept it…
See also “Mapping Music” from Harvard Magazine , Jan.-Feb. 2007—
“Life inside an orbifold is a non-Euclidean world”
— as well as the cover story “The Shape of Music” from Princeton Alumni Weekly ,
Feb. 9, 2011, and “Bead Game” + music in this journal (click, then scroll down).
Those impressed by the phrase “non-Euclidean” may also enjoy
Non-Euclidean Blocks and Pilate Goes to Kindergarten.
The “Bead Game” + music search above includes, notably, a passage describing a
sort of non-Euclidean abacus in the classic 1943 story “Mimsy Were the Borogoves.”
For a visually related experience, see the video “Chord Geometries Demo: Chopin
on a Mobius Strip” at a music.princeton.edu web page.
* Motto of the American Mathematical Society, said to be also the motto of Plato’s Academy.
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Thursday, February 24, 2011
Now Lens (continued)
"…the farther back we manage to wiggle
the more we need the loan of a lens…." —Finnegans Wake
For some background on the lens below, see the sermon on February 20th, 2011.
“The quad gospellers may own the targum but any of the Zingari shoolerim
may pick a peck of kindlings yet from the sack of auld hensyne.”
The above "Zingari shoolerim" passage was quoted here in Frame Tales (November 10th, 2008).
That post concerned the Heinrich Zimmer tale "The King and the Corpse." Some related material—
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Sunday, November 28, 2010
Permanent
A counterbalance to this morning's New York Times story (see previous post) on a leftist Columbia University seminar might be C.S. Lewis's famous quote "It's all in Plato." Unfortunately, a search for discussions of this quote yields, as the top result, a typically shoddy Christian polemic.
From a Christian professor at Seattle Pacific University—
"Friedrich Nietzsche (1844-1900) can serve us well as a negative example. Nietzsche's 'new morality' was 'mere innovation,' according to [C.S.] Lewis [Christian Reflections ]. Nietzsche announced through Zarathustra (and numerous other ways) the relativity of all values and moralities, saying that each people had adhered to a different scheme of values worked out solely in connection with local conditions. 'Nothing is true, everything is permitted,' states Nietzsche, quoting one of Dostoevsky's characters."
No source is given for either the Nietzsche quote or the alleged Dostoevky quote.
This is from a web page titled "In Defense of the Permanent Things." The Christian professor is, according to a far better scholar, permanently wrong. See Note 8 on pages 586-587 of Walter Kaufmann's Basic Writings of Nietzsche (Random House, November 28, 2000).
One hopes that a more capable scholar, such as Lewis himself* was, might at some point attack the Columbia University leftist nonsense— and Nietzsche— by quoting a more damning passage, such as
" 'Everything is false! Everything is permitted!' "
(from Kaufmann's edition of The Will to Power , Random House, 1968, page 326)
This version of the "Everything is permitted" quote is much more directly related to Nietzsche's relativism, as seen in this image of Kaufmann's edition—
* Or Steven Michels of Sacred Heart University in Fairfield, CT.
See his "Nietzsche, Nihilism, and the Virtue of Nature"— in particular, note 29.
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Saturday, October 30, 2010
Keanu vs. the Devil
(Continued from Little Buddha (1994), The Matrix (1999), and Constantine (2005))
This post was suggested by yesterday's post on Habermas and by his 1962 book The Structural Transformation of the Public Sphere (English translation, 1989).
The "public sphere" of Habermas has come to pass; it is, of course, the World-Wide Web.
For October 30, the day leading up to Devil's Night, a more private sphere—though in a public setting— seems appropriate…
The Day the Earth Stood Still (2008)
A Keanu Reeves scene related to this image—
"The low point of the movie’s persuasiveness is the single scene with Professor Barnhardt (John Cleese) — in the original an Einstein-like scientist who impresses Klaatu with his highly evolved thinking, here a caricature of professorial enlightenment. Helen decides to bring Klaatu to Professor Barnhardt when Klaatu professes his disappointment with earth’s leaders. 'Those aren’t our leaders!' she protests earnestly. 'Let me take you to one of our leaders!'"
A perhaps more persuasive scene, from today's New York Times—
Prize in Hand, He Keeps His Eye on Teaching
Nobel winner Mario Vargas Llosa teaches
a seminar on Borges at Princeton
(Photo by James Leynse for The New York Times )
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Tuesday, June 29, 2010
Kind of Bleu
Click for a Global Partners story.
Related material:
"We want to promote the vitality of mathematics
by playing an increasingly active role in political affairs."
— Princeton Class Notes, Jan. 27, 1999, remark by Felix E. Browder,
then president-elect of the American Mathematical Society (AMS).
See also web pages on Browder's brothers
William (also an AMS president, 1989-1990)
and Andrew and their father Earl.
Earl was General Secretary of
the Communist Party USA from 1930 to 1944.
Princeton Class Notes on the Browders— "The senior Browder 'discouraged me and my two brothers from taking an active part in politics, but strongly encouraged our intellectual interests.' That all three brothers became mathematicians– the others are Princeton professor William Browder '58 (a former president of the AMS) and Brown professor Andrew Browder– is an outcome for which Felix Browder 'can offer no rational explanation.'"
"As a trusted partner, we do more than consult and train.
We add a new dimension to our client’s thinking…."
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Monday, June 21, 2010
Test
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the four-diamond figure
with the Pythagorean theorem.
A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
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Monday, June 14, 2010
Theory of Ambiguity
Théorie de l'Ambiguité
According to a 2008 paper by Yves André of the École Normale Supérieure of Paris—
"Ambiguity theory was the name which Galois used
when he referred to his own theory and its future developments."
The phrase "the theory of ambiguity" occurs in the testamentary letter Galois wrote to a friend, Auguste Chevalier, on the night before Galois was shot in a duel.
Hermann Weyl in Symmetry, Princeton University Press, 1952—
"This letter, if judged by the novelty and profundity of ideas it contains, is perhaps
the most substantial piece of writing in the whole literature of mankind."
Conclusion of the Galois testamentary letter, according to
the 1897 Paris edition of Galois's collected works—
The original—
A transcription—
Évariste GALOIS, Lettre-testament, adressée à Auguste Chevalier—
Tu sais mon cher Auguste, que ces sujets ne sont pas les seuls que j'aie
explorés. Mes principales méditations, depuis quelques temps,
étaient dirigées sur l'application à l'analyse transcendante de la théorie de
l'ambiguité. Il s'agissait de voir a priori, dans une relation entre des quantités
ou fonctions transcendantes, quels échanges on pouvait faire, quelles
quantités on pouvait substituer aux quantités données, sans que la relation
put cesser d'avoir lieu. Cela fait reconnaitre de suite l'impossibilité de beaucoup
d'expressions que l'on pourrait chercher. Mais je n'ai pas le temps, et mes idées
ne sont pas encore bien développées sur ce terrain, qui est
immense.
Tu feras imprimer cette lettre dans la Revue encyclopédique.
Je me suis souvent hasardé dans ma vie à avancer des propositions dont je n'étais
pas sûr. Mais tout ce que j'ai écrit là est depuis bientôt un an dans ma
tête, et il est trop de mon intérêt de ne pas me tromper pour qu'on
me soupconne d'avoir énoncé des théorèmes dont je n'aurais pas la démonstration
complète.
Tu prieras publiquement Jacobi et Gauss de donner leur avis,
non sur la vérité, mais sur l'importance des théorèmes.
Après cela, il y aura, j'espère, des gens qui trouveront leur profit
à déchiffrer tout ce gachis.
Je t'embrasse avec effusion.
E. Galois Le 29 Mai 1832
A translation by Dr. Louis Weisner, Hunter College of the City of New York, from A Source Book in Mathematics, by David Eugene Smith, Dover Publications, 1959–
You know, my dear Auguste, that these subjects are not the only ones I have explored. My reflections, for some time, have been directed principally to the application of the theory of ambiguity to transcendental analysis. It is desired see a priori in a relation among quantities or transcendental functions, what transformations one may make, what quantities one may substitute for the given quantities, without the relation ceasing to be valid. This enables us to recognize at once the impossibility of many expressions which we might seek. But I have no time, and my ideas are not developed in this field, which is immense.
Print this letter in the Revue Encyclopédique.
I have often in my life ventured to advance propositions of which I was uncertain; but all that I have written here has been in my head nearly a year, and it is too much to my interest not to deceive myself that I have been suspected of announcing theorems of which I had not the complete demonstration.
Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of the theorems.
Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess.
J t'embrasse avec effusion.
E. Galois. May 29, 1832.
Translation, in part, in The Unravelers: Mathematical Snapshots, by Jean Francois Dars, Annick Lesne, and Anne Papillaut (A.K. Peters, 2008)–
"You know, dear Auguste, that these subjects are not the only ones I have explored. For some time my main meditations have been directed on the application to transcendental analysis of the theory of ambiguity. The aim was to see in a relation between quantities or transcendental functions, what exchanges we could make, what quantities could be substituted to the given quantities without the relation ceasing to take place. In that way we see immediately that many expressions that we might look for are impossible. But I don't have the time and my ideas are not yet developed enough in this vast field."
Another translation, by James Dolan at the n-Category Café—
"My principal meditations for some time have been directed towards the application of the theory of ambiguity to transcendental analysis. It was a question of seeing a priori in a relation between quantities or transcendent functions, what exchanges one could make, which quantities one could substitute for the given quantities without the original relation ceasing to hold. That immediately made clear the impossibility of finding many expressions that one could look for. But I do not have time and my ideas are not yet well developed on this ground which is immense."
Related material—
"Renormalisation et Ambiguité Galoisienne," by Alain Connes, 2004
"La Théorie de l’Ambiguïté : De Galois aux Systèmes Dynamiques," by Jean-Pierre Ramis, 2006
"Ambiguity Theory, Old and New," preprint by Yves André, May 16, 2008,
"Ambiguity Theory," post by David Corfield at the n-Category Café, May 19, 2008
"Measuring Ambiguity," inaugural lecture at Utrecht University by Gunther Cornelissen, Jan. 16, 2009
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Tuesday, June 1, 2010
Contra Harvard
Today is commencement day at Princeton.
Sunday's A Post for Galois was suggested, in part, by the fact that the founder and CEO of Amazon.com was that day's Princeton baccalaureate speaker. The Galois post linked to the Amazon reviews of one Christopher G. Robinson, a resident of Cambridge, Mass., whose Amazon book list titled "Step Right Up!" reflects a continuing libertine tradition at Harvard.
For Princeton's commencement day, it seems fitting to cite another Amazon document that reflects the more conservative values of that university.
I recommend the review Postmodern Pythagoras, by Matthew Milliner. Milliner is, in his own words, "an art history Ph.D. candidate at Princeton University."
See also Milliner's other reviews at Amazon.com.
"For every kind of libertine,
there is a kind of cross."
— Saying adapted from Pynchon
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Sunday, May 2, 2010
Legal Fiction–
Murphy's Last Stand
In memory of Walter F. Murphy, a leading constitutional scholar and McCormick Professor of Jurisprudence at Princeton from 1968 to 1995. Murphy was also the author of a bestselling 1979 novel, The Vicar of Christ. He died at 80 on Tuesday, April 20, 2010, quod vide.
His novel, according to this morning's New York Times, "tells the story of an American who fights valiantly in the Korean War (as Professor Murphy did), becomes chief justice of the United States, resigns to become a monk and is eventually elected the first American pope." An eventful tale.
For a good review of Murphy's novel, see "The Doomed Hero." This review, and yesterday's Log24 Law Day post, which mentions the concept of "the mighty music of the innermost heaven," suggest revisiting a Log24 post of August 28, 2009 and a hymn by Brian Wilson—
 Send "In My Room" Ringtone to Cell
There's a world where I can go In this world I lock out Do my dreaming and my scheming lie awake and pray Now it's dark and I'm alone |
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Wednesday, January 6, 2010
Epiphany Revisited
|
January 06, 2007 Picture of Nothing
“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….” Related material: The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.” — Tom Wolfe, The Painted Word “Concept (scholastics’ verbum mentis)– theological analogy of Son’s procession as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957, second printing 1963, page 162
“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”
— Jim Holt, Big-Bang Theology, from Slate‘s “High Concept” department  “Bang.” “…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….” For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.
|
Some Context:
See also Nativity.
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Monday, September 7, 2009
Monday September 7, 2009
Magic Boxes
"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"
— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."
Commentary
on the passage:
Part I "The Magic Box," shown on Turner Classic Movies earlier tonight
Part II: "Mimsy Were the Borogoves," a classic science fiction story:
"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."
Part III: A Crystal Block —
Image of pencils is by
Diane Robertson Design.
Related material:
"A Four-Color Theorem."
Part IV:
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Friday, September 4, 2009
Friday September 4, 2009
Closing the Circle
Continued from Monday
“This is a chapel
of mischance;
ill luck betide it, ’tis
the cursedest kirk
that ever I came in!”
Philip Kennicott on
Kirk Varnedoe in
The Washington Post:
“Varnedoe’s lectures were
ultimately about faith,
about his faith in
the power of abstraction,
and abstraction as a kind of
anti-religious faith in itself….”
Kennicott’s remarks were
on Sunday, May 18, 2003.
They were subtitled
“Closing the Circle
on Abstract Art.”
Also on Sunday, May 18, 2003:
“Will the circle be unbroken?
As if some southern congregation
is praying we will come to understand.”
Princeton University Press:
See also
Parmiggiani’s
Giordano Bruno —
Dürer’s Melencolia I —
and Log24 entries
of May 19-22, 2009,
ending with
“Steiner System” —
George Steiner on chess
(see yesterday morning):
“There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything– marriage, mortgages, careers, the Russian Revolution– in order to spend their days and nights moving little carved objects up and down a quadrate board.”
“Allegoric associations of death with chess are perennial….”
Yes, they are.
April is Math Awareness Month.
This year’s theme is “mathematics and art.”
Cf. both of yesterday’s entries.
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Monday, August 17, 2009
Monday August 17, 2009
Design Theory,
continued
continued
“… Kirkman has established an incontestable claim to be regarded as the founding father of the theory of designs.”
— “T.P. Kirkman, Mathematician,” by N.L. Biggs, Bulletin of the London Mathematical Society, Volume 13, Number 2 (March 1981), 97-120.
This paper is now available online for $12.
For more about this subject, see Design Theory, by Beth, Jungnickel, and Lenz, Cambridge U. Press, Volume I (2nd ed., 1999, 1120 pages) and Volume II (2nd ed., 2000, 513 pages).
For an apparently unrelated subject with the same name, see Graphic Design Theory: Readings from the Field, by Helen Armstrong (Princeton Architectural Press, 2009).
For what the two subjects have in common, see Block Designs in Art and Mathematics.
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