Log24

The phrase “the permutation group S_n” refers to a
particular  group of permutations that act on an
n -element set N— namely, all  of them. For a given n ,
there are, in general, many  permutation groups that
act on N.  All but one are smaller than S_n .

In other words, the phrase “the permutation group S_n”
does not  imply that “S_n ” is a symbol for a structure
associated with n  called “the  permutation group.”
It is instead a symbol for “the symmetric  group,” the largest
of (in general) many permutation groups that act on N.

This point seems to have escaped John Baez.

For two misuses by Baez of the phrase “permutation group” at the
n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

There is  such a thing as geometry.*

* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.

(Continued)

"There is  such a thing as a tesseract."

— Saying from Crosswicks

See also March 5, 2011.

Adapted from the above passage —

"So did L'Engle understand four-dimensional geometry?"

No and Yes.

For readers of The Daily Princetonian :

(From a site advertised in the
Princetonian  on March 11, 2013)

For readers of The Harvard Crimson :

For some background, see Crimson Easter Egg and the Diamond 16 Puzzle.

For some (very loosely) related narrative, see Crosswicks in this journal
and the Crosswicks Curse  in a new novel by Joyce Carol Oates.

"There is  such a thing as a tesseract."
— Crosswicks author Madeleine L'Engle

Mathematics and Narrative, continued

"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"

— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th

"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."

— Dan Conover on September 1st, 2011

"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'

I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."

— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996

"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"

— Stanford Encyclopedia of Philosophy

"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."

— Interviewer for the National Association of Scholars

From that conversation—

Wolfe : "People in academia should start insisting on objective scholarship, insisting  on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"

As for "mis-imagined quantum pins"…
This  journal on the date of the above interview— February 28, 2008—

Illustration from a Perimeter Institute talk given on July 20, 2005

The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by  Conover, "The Last Epiphany," and four posts from September 1st, 2011—

Boundary,  How It Works,  For Thor's Day,  and The Galois Tesseract.

Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.

"There is  such a thing as a tesseract." —A Wrinkle in Time

Dies Natalis of
Emil Artin

From the September 1953 Bulletin of the American Mathematical Society—

Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–

"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."

Art Versus Chaos

From an exhibit,
"Reimagining Space"

The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."

For related mathematical details see
Diamond Theory in 1937.

"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
— Fritz Leiber, 1959

See also the Log24 entry for
Nov. 26,  2009, the date that
Forakis died.

"There is such a thing
as a tesseract."
— Madeleine L'Engle, 1962

Geometry of the 4×4 Square:

http://log24.com/theory/geometry.html

“There is such a thing as a tesseract.”
— A Wrinkle in Time

Aptheker

A Look at the Rat

In memory of Herbert Aptheker, theoretician of the American Communist Party, who died on St. Patrick’s Day, 2003 —

From The New Yorker, issue dated March 24, 2003, Louis Menand on Edmund Wilson’s To the Finland Station:

“Wilson did know what was going on in the Soviet Union in the nineteen-thirties, as his pages on Stalin in To the Finland Station make clear. The problem wasn’t with Stalin; the problem was with Lenin, the book’s ideal type of the intellectual as man of action. Wilson admitted that he had relied on publications controlled by the Party for his portrait of Lenin. (Critical accounts were available; for example, the English translation of the émigré Mark Landau-Aldanov’s Lenin was published, by Dutton, in 1922.) Lenin could create an impression of selfless humanitarianism; he was also a savage and ruthless politician—a ‘pail of milk of human kindness with a dead rat at the bottom,’ as Vladimir Nabokov put it to Wilson in 1940, after reading To the Finland Station.  In the introduction to the 1972 edition, Wilson provided a look at the rat. He did not go on to explain in that introduction that the most notorious features of Stalin’s regime—the use of terror, the show trials, and the concentration camps—had all been inaugurated by Lenin. To the Finland Station begins with Napoleon’s betrayal of the principles of the French Revolution; it should have ended with Lenin’s betrayal of European socialism.” 

From Herbert Aptheker, “More Comments on Howard Fast“:

“We observe that in the list of teachers whom Howard Fast names as most influential in his own life there occur the names of fourteen individuals from Jefferson to Bernard Shaw, Upton Sinclair to Marx, Douglass to Engels, but there is no room for Lenin.
   He is, I think, an important teacher, too; indeed, in my view, Lenin is the greatest figure in the whole galaxy of world revolutionary leaders. He is, certainly, the greatest analyzer of and fighter against imperialism.”

For more on Howard Fast, see my entry
“Death Knell” of March 13, 2003. 

For a look at the pail of milk, see
the New Yorker cover in Geometry for Jews.

For a more cheerful look at geometry
on this St. Joseph’s Day, see
Harry J. Smith’s

Tesseract Site.

“There is such a thing as a tesseract.”
— A Wrinkle in Time