From a post of August 13, "The Divided Square" —
From a post of April 25, 2016, "Peirce's Accounts of the Universe" —
For the Peirce arrangement, see the imaginary film
reviewed in last night's post Packend! .
This post was suggested by this morning's New York Times story on the missing cornerstone of St. Patrick's Cathedral and by the recent design for an official T-shirt celebrating Harvard's 375th anniversary—
In Harvard's case, the missing piece beneath the cathedral-like spire* is the VERITAS on the college shield.
Possible sources for a shield image representing VERITAS—
1. "Patrick Blackburn" in this journal, which might be combined with
2. Reflections on Kurt Gödel ** by Hao Wang, Chapter 9, "To Fit All the Parts Together"—
"The metaphor of fitting parts together readily suggests
the concrete image of solving a picture puzzle…." (p. 243)
Or the image of a Wang tiles puzzle.
A graphic image, colorful but garish, that summarizes these two sources—
Shield with matching Wang tiles
* The Lowell House bell tower
** MIT Press, first published in 1987
Wednesday March 10, 2004 — m759 @ 4:07 AM “Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. He [Mallarmé] became aware, in Millan’s* words, ‘of the extremely fine line
separating absence and presence, being and nothingness, life and death, which later … he could place at the very centre of his work and make the cornerstone of his personal philosophy and his mature poetics.’ “ — John Simon, "Squaring the Circle"
* A Throw of the Dice: The Life of Stéphane Mallarmé , |
See also Cornerstone.
Not-so-mature poetics —
… and completely im mature poetics —
See as well other posts now tagged Taiji , a search for Chinese Checkers,
and a recent Harvard Crimson piece by Gish Jen.
Critic John Simon reportedly died at 94 on Sunday in Valhalla, N.Y. —
A search for Simon in this journal yields …
Wednesday March 10, 2004 — m759 @ 4:07 AM “Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. He [Mallarmé] became aware, in Millan’s* words, ‘of the extremely fine line
separating absence and presence, being and nothingness, life and death, which later … he could place at the very centre of his work and make the cornerstone of his personal philosophy and his mature poetics.’ “ — John Simon, "Squaring the Circle" * A Throw of the Dice: The Life of Stéphane Mallarmé , by Gordon Millan |
See also Cornerstone.
See also "Cornerstone" in this journal and …
A sidebar from a Google search today —
This suggests a review of posts now tagged Obelisk,
which include …
For Scarlett
From a search for "Preparation" in this journal —
"In a nutshell, the book serves as an introduction to
Gauss' theory of quadratic forms and their composition laws
(the cornerstone of his Disquisitiones Arithmeticae ) from the
modern point of view (ideals in quadratic number fields)."
From a film in which Scarlett portrays a goddess —
Madness related to several recent posts —
Then, with an unheard splash which sent from the silver water to the shore a line of ripples echoed in fear by my heart, a swimming thing emerged beyond the breakers. The figure may have been that of a dog, a human being, or something more strange. It could not have known that I watched—perhaps it did not care—but like a distorted fish it swam across the mirrored stars and dived beneath the surface. After a moment it came up again, and this time, since it was closer, I saw that it was carrying something across its shoulder. I knew, then, that it could be no animal, and that it was a man or something like a man, which came toward the land from a dark ocean. But it swam with a horrible ease.
— From "The Night Ocean," by H. P. Lovecraft |
Related news —
"When hard-liners seized power in Moscow in August 1991
and imprisoned Mr. Gorbachev in his vacation house on the
Black Sea, Mr. Chernyaev, a guest there and a powerful swimmer,
offered to smuggle out a note by swimming to a beach more than
three miles away. Uncertain where he could take the note, they
dropped the plan. The coup quickly failed in any case."
"Backstories do more than amuse guests.
They anchor the hosts.
It's their cornerstone.
The rest of their identity is built around it, layer by layer."
— Elsie Hughes in "Westworld," Season 1, Episode 3,
"The Stray," at 30:09
See also cornerstone in the Bible.
Dr. Robert Ford (Anthony Hopkins) earlier in that same episode —
The American Mathematical Society yesterday:
Harvey Cohn (1923-2014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."
See also Symbology, The Lost Cornerstone, and Now, Here's My Plan.
Presidential candidate Mitt Romney on Monday, April 2, 2012—
"I think there is in this country a war on religion.
I think there is a desire to establish a religion
in America known as secularism."
Nancy Haught of The Oregonian on Leap Day, Feb. 29, 2012—
William Hamilton, the retired theologian who declared in the 1960s that God was dead, died Tuesday [Feb. 28, 2012] in his downtown Portland apartment at 87. Hamilton said he'd been haunted by questions about God since he was a teenager. Years later, when his conclusion was published in the April 8, 1966, edition of Time Magazine, he found himself in a hornet's nest. Time christened the new movement "radical theology" and Hamilton, one of its key figures, received death threats and inspired angry letters to the editor in newspapers that carried the story. He encountered hostility at Colgate Rochester Crozer Divinity School, where he had been teaching theology, and lost his endowed chair in 1967. Hamilton moved on to teach religion at New College in Sarasota, Fla. |
(See also this journal on Leap Day.)
From New College: The Honors College of Florida—
Oct. 11, 1960: New College is founded as a private college 1961: Trustees obtain options to purchase the former Charles Ringling estate on Sarasota Bay and 12 acres of airport land facing U.S. 41 held by private interests. The two pieces form the heart of the campus Nov. 18, 1962: the campus is dedicated. Earth from Harvard is mixed with soil from New College as a symbol of the shared lofty ideals of the two institutions. |
See also, in this journal, "Greatest Show on Earth" and The Harvard Crimson—
The Harvard Crimson, Online Edition |
Sunday, Oct. 8, 2006 |
POMP AND
Friday, Oct. 6:
The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden. |
OPINION At Last, a By THE CRIMSON STAFF The Trouble By SAHIL K. MAHTANI |
"Plan 9 is an operating system kernel but also a collection of accompanying software."
— Webpage pointed out by the late Dennis Ritchie,
father of the programming language C
and co-developer of Unix, who reportedly died on October 8.
From Ritchie's own home page—
"A brief biography, in first person instead of obituary style."
From that biography—
"Today, as a manager of a small group of researchers, I promote exploration of distributed operating systems, languages, and routing/switching hardware. The recent accomplishments of this group include the Plan 9 operating system…."
Another operating system is that of Alfred Bester.
My laptop now includes his classic The Stars My Destination ,
downloaded this morning…
Not much compared to Widener Library (see this morning's Lost Cornerstone),
but sufficient for present purposes…
"Simple jaunt." — "The Comedian as the Letter C"
See also Plan 9 from Outer Space in this journal.
From this journal:
Friday December 5, 2008Mirror-Play of
the Fourfold For an excellent commentary View selected pages Play and the Aesthetic Dimension (Mihai I. Spariosu, Related material: – and Theme and Variations. |
Transition to the
Garden of Forking Paths–
(See For Baron Samedi)–
The Found Symbol
and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–
Pages 354-355
On the mirror-play of the fourfold
Pages 356-357
Shaking up a whole culture
Pages 358-359
Cornerstone and crossroads
Pages 360-361
A deep impression embedded in stone
Pages 362-363
A certain Y, a certain V
Pages 364-365
The world is Zeus's play
Page 366
It was necessary to begin again
“The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast.”
Natasha Wescoat, 2004 Shakespearean Fool |
|
Not to mention Euclid and Picasso | |
(Click on pictures for details. Euclid is represented by Alexander Bogomolny, Picasso by Robert Foote.)
|
See also works by the late Arthur Loeb of Harvard’s Department of Visual and Environmental Studies.
“I don’t want to be a product of my environment. I want my environment to be a product of me.” — Frank Costello in The Departed
For more on the Harvard environment,
see today’s online Crimson:
The Harvard Crimson, Online Edition |
Sunday, Oct. 8, 2006 |
POMP AND Friday, Oct. 6: The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden. |
OPINION At Last, a By THE CRIMSON STAFF The Trouble By SAHIL K. MAHTANI |
— Dean G. Hoffman, Auburn U.,
July 2001 Rutgers talk
Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):
Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).
From a March 10, 2004, entry:
“Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. [Mallarmé] became aware, in Millan’s* words, ‘of the extremely fine line
separating absence and presence, being and nothingness, life and death, which
— John Simon, Squaring the Circle
* A Throw of the Dice: The Life of Stéphane Mallarmé, by Gordon Millan
For those who prefer
art that is more lurid:
(Photo in lower half
from Cinetribulations)
Related material:
The Line
From a March 10, 2004, entry:
“Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. [Mallarmé] became aware, in Millan’s* words, ‘of the extremely fine line separating absence and presence, being and nothingness, life and death, which — John Simon, Squaring the Circle * A Throw of the Dice: The Life of Stéphane Mallarmé, by Gordon Millan The illustration of the “fine line” is not by Mallarmé but by myself. (See Songs for Shakespeare, March 5, where the line separates being from nothingness, and Ridgepole, March 7, where the line represents the “great primal beginning” of Chinese philosophy (or, equivalently, Stevens’s “first idea” or Mallarmé’s line “separating absence and presence, being and nothingness, life and death.”) |
By the Associated Press,
Saturday, March 13, 2004:
“Dave Schulthise, known as Dave Blood during his career as a bassist with the 1980’s Philadelphia punk-rock band the Dead Milkmen, died on Wednesday [March 10, 2004] at the home of friends in North Salem, N.Y. He was 47.
‘David chose to end his life,’ Mr. Schulthise’s sister, Kathy, wrote on the band’s Web site.”
I walk the thinnest line
I walk the thinnest line
I walk the thinnest line
Between the light and dark sides of my mind
— The Dead Milkmen, Beelzebubba album
Related material: The Word in the Desert.
Ennui of the First Idea
The ennui of apartments described by Stevens in "Notes Toward a Supreme Fiction" (see previous entry) did not, of course, refer to the "apartments" of incidence geometry. A more likely connection is with the apartments — the "ever fancier apartments and
"Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. He [Mallarmé] became aware, in Millan’s* words, 'of the extremely fine line
separating absence and presence, being and nothingness, life and death, which
— John Simon, Squaring the Circle
* A Throw of the Dice: The Life of Stéphane Mallarmé, by Gordon Millan
The illustration of the "fine line" is not by Mallarmé but by myself. (See Songs for Shakespeare, March 5, where the line separates being from nothingness, and Ridgepole, March 7, where the line represents the "great primal beginning" of Chinese philosophy (or, equivalently, Stevens's "first idea" or Mallarmé's line "separating absence and presence, being and nothingness, life and death.")
Theology for a Small Planet
THE HARVARD DIVINITY BULLETIN for Fall 1989 contained a special section, "Theology for a Small Planet," with a number of short articles by divinity school faculty and others addressing environment and theology.
From The Harvard Divinity Bulletin, XIX, 3 (1989):
" While Angels Weep…"
Doing Theology on a Small Planet
Timothy C. Weiskel
© Copyright, 1989, Timothy C. Weiskel
…We continue to strut and prance about with a sense of supreme self-importance as if all creation were put in place for our benefit….
From where does such arrogance come? How can our beliefs be so far out of touch with our knowledge? How can we maintain such an inflated sense of personal, collective and species self-importance? ….
The answer, in part, is that Western religious traditions have generated and sustained this petty arrogance….
Western cultures have come to believe religiously in their own power, importance and capacity to dominate and control nature.
Some religious groups have transcribed and elaborated creation myths which serve to ennoble and authorize this illusion of domination. In these myths a supreme and omnipotent God figure (usually portrayed as male) is said to have created humankind and enjoined this species to be "fruitful and multiply" and "subdue" the earth. Moreover, it is often a feature of these traditions that selected human groups come to feel entitled, empowered or specially ordained by such a God to be his "chosen people." Through their actions and history, it is believed, this God allegedly manifested his intent for the planet as a whole. In short, human groups created God in their own image and generated divine narratives that accorded themselves privileged status in the whole of creation….
…science itself has become the cornerstone of modern mankind's religiously held belief in human control. In our era, this kind of arrogant science, like the self-important religious traditions of the past, must be questioned….
In short, we all stand in need of a theology for a small planet.
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