(Continued from November 27, 2010.)
The previous post suggests a review of a Tilman Piesk
illustration, with the general form of a 4-simplex, from
the Wikipedia article titled Simplex . As the article
notes, the lines shown connecting points are those of a
tesseract.
An Adamantine View of "The [Philosophers'] Stone"
The New York Times column "The Stone" on Sunday, Nov. 21 had this—
"Wittgenstein was formally presenting his Tractatus Logico-Philosophicus , an already well-known work he had written in 1921, as his doctoral thesis. Russell and Moore were respectfully suggesting that they didn’t quite understand proposition 5.4541 when they were abruptly cut off by the irritable Wittgenstein. 'I don’t expect you to understand!' (I am relying on local legend here….)"
Proposition 5.4541*—
Related material, found during a further search—
A commentary on "simplex sigillum veri" leads to the phrase "adamantine crystalline structure of logic"—
For related metaphors, see The Diamond Cube, Design Cube 2x2x2, and A Simple Reflection Group of Order 168.
Here Łukasiewicz's phrase "the hardest of materials" apparently suggested the commentators' adjective "adamantine." The word "diamond" in the links above refers of course not to a material, but to a geometric form, the equiangular rhombus. For a connection of this sort of geometry with logic, see The Diamond Theorem and The Geometry of Logic.
For more about God, a Stone, logic, and cubes, see Tale (Nov. 23).
* 5.4541 in the German original—
Die Lösungen der logischen Probleme müssen einfach sein,
denn sie setzen den Standard der Einfachheit.
Die Menschen haben immer geahnt, dass es
ein Gebiet von Fragen geben müsse, deren Antworten—
a priori—symmetrisch, und zu einem abgeschlossenen,
regelmäßigen Gebilde vereint liegen.
Ein Gebiet, in dem der Satz gilt: simplex sigillum veri.
Here "einfach" means "simple," not "neat," and "Gebiet" means
"area, region, field, realm," not (except metaphorically) "sphere."
(Title purloined from Marie-Louise von Franz.)
BCE/CE — A game with three letters … See Michener Game.
BC/AD — A game with four letters … See Eddington Game.
ABCDE — A game with five letters … See Simplex.
From other posts tagged Tetrahedron vs. Square —
A Scholium for Chomsky —
The ABC of words —
A nutshell —
Mathematics:
From Log24 "Pyramid Game" posts —
The letter labels, but not the tetrahedron, are from Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906), page 13.
Narrative:
The New York Times yesterday ("2022-05-24T21:54:19.000Z")
on a Saturday, May 21, death —
"Colin Cantwell, an animator, conceptual artist and computer expert
who played significant production roles in seminal science fiction films
like '2001: A Space Odyssey,' 'Star Wars' and 'WarGames,' died
on May 21 at his home in Colorado Springs, Colo. He was 90."
Cantwell at Teotihuacan pyramid, September 26, 2019 —
A different image, also from September 26, 2019,
in other Log24 posts tagged Pyramid Game —
The letter labels, but not the tetrahedron, are from Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906),
page 13.
"… out of all things there comes a unity,
and out of a unity all things . . . . "
— Heraclitus, according to de Beer quoting McKirahan
An image we may regard as illustrating
the group-identity symbol "e" for "Einheit " —
From other posts tagged Tetrahedron vs. Square —
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
Illustration (central detail a from the above tetrahedral figure) —
A Harvard Variation
from Timothy Leary —
The topics of Harvard and Leary suggest some other cultural
history, from The Coasters — "Poison Ivy" and "Yakety Yak."
(The title refers to Log24 posts now tagged Fire Temple.)
In memory of a New Yorker cartoonist who
reportedly died at 97 on October 3, 2019 …
"Read something that means something."
— New Yorker advertising slogan
From posts tagged Tetrahedron vs. Square —
This journal on October 3 —
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
Illustration (central detail a from the above tetrahedral figure) —
(A sequel to Simplex Sigillum Veri and
Rabbit Hole Meets Memory Hole)
” Wittgenstein does not, however, relegate all that is not inside the bounds
of sense to oblivion. He makes a distinction between saying and showing
which is made to do additional crucial work. ‘What can be shown cannot
be said,’ that is, what cannot be formulated in sayable (sensical)
propositions can only be shown. This applies, for example, to the logical
form of the world, the pictorial form, etc., which show themselves in the
form of (contingent) propositions, in the symbolism, and in logical
propositions. Even the unsayable (metaphysical, ethical, aesthetic)
propositions of philosophy belong in this group — which Wittgenstein
finally describes as ‘things that cannot be put into words. They make
themselves manifest. They are what is mystical’ ” (Tractatus 6.522).
— Stanford Encyclopedia of Philosophy , “Ludwig Wittgenstein”
| From Tractatus Logico-Philosophicus by Ludwig Wittgenstein.
(First published in Annalen der Naturphilosophie ,1921. 5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined — a priori — to form a self-contained system. A realm subject to the law: Simplex sigillum veri. |
Somehow, the old Harvard seal, with its motto “Christo et Ecclesiae ,”
was deleted from a bookplate in an archived Harvard copy of Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906).
In accordance with Wittgenstein’s remarks above, here is a new
bookplate seal for Whitehead, based on a simplex —
The 15 points of the finite projective 3-space PG(3,2)
arranged in tetrahedral form:
The letter labels, but not the tetrahedral form,
are from The Axioms of Projective Geometry , by
Alfred North Whitehead (Cambridge U. Press, 1906).
The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."
Screen Rant on July 31, 2019:
A Google Search sidebar this morning:
Apocalypse Soon! —
Illustration from other posts now tagged
Cervantes Meets Pascal.
See as well Verhexung in this journal.
This post was suggested by . . .
|
arXiv.org > quant-ph > arXiv:1905.06914 Quantum Physics Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones J. P. Marceaux, A. R. P. Rau (Submitted on 14 May 2019) A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie-Clifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral three-dimensional simplex of 15 points are key objects that feature in this study. Comments:16 pages, 10 figures Subjects:Quantum Physics (quant-ph) Cite as:arXiv:1905.06914 [quant-ph] (or arXiv:1905.06914v1 [quant-ph] for this version) Submission history
From: A. R. P. Rau [view email] |
See also other posts tagged Tetrahedron vs. Square.
The phrase "twin pillars" in a New York Times Fashion & Style
article today suggests a look at another pair of pillars —
This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as "parodies of ideas (which are ghosts)." (See the previous post.)
For a relationship between a 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.
Last midnight's post quoted poet John Hollander
on Cervantes—
"… the Don’s view of the world is correct at midnight,
and Sancho’s at noon."
The post concluded with a figure that might, if
rotated slightly, be regarded as a sort of Star of
David or Solomon's Seal. The figure's six vertices
may be viewed as an illustration of Pascal's
"mystic hexagram."
Pacal's hexagram is usually described
as a hexagon inscribed in a conic
(such as a circle). Clearly the hexagon
above may be so inscribed.
The figure suggests that last midnight's Don be
played by the nineteenth-century mathematician
James Joseph Sylvester. His 1854 remarks on
the nature of geometry describe a different approach
to the Pascal hexagram—
| "… the celebrated theorem of Pascal known under the name of the Mystic Hexagram, which is, that if you take two straight lines in a plane, and draw at random other straight lines traversing in a zigzag fashion between them, from A in the first to B in the second, from B in the second to C in the first, from C in the first to D in the second, from D in the second to E in the first, from E in the first to F in the second and finally from F in the second back again to A the starting point in the first, so as to obtain ABCDEF a twisted hexagon, or sort of cat's-cradle figure and if you arrange the six lines so drawn symmetrically in three couples: viz. the 1st and 4th in one couple, the 2nd and 5th in a second couple, the 3rd and 6th in a third couple; then (no matter how the points ACE have been selected upon one of the given lines, and BDF upon the other) the three points through which these three couples of lines respectively pass, or to which they converge (as the case may be) will lie all in one and the same straight line." |
For a Sancho view of Sylvester's "cat's cradle," see some twentieth-century
remarks on "the most important configuration of all geometry"—
"Now look, your grace," said Sancho,
"what you see over there aren't giants,
but windmills, and what seems to be arms
are just their sails, that go around in the wind
and turn the millstone."
"Obviously," replied Don Quijote,
"you don't know much about adventures.”
From a 2003 interview by Paul Devlin (PD) with poet John Hollander (JH),
who reportedly died Saturday—
|
PD: You wrote in the introduction to the new edition of Reflections on Espionage that whenever you have been "free of political callowness" it was partly as a result of reading W.H. Auden, George Orwell, and George Bernard Shaw. Do you think these writers might possibly be an antidote to political callowness that exists in much contemporary literary criticism? JH: If not they, then some other writers who can help one develop within one a skepticism strongly intertwined with passion, so that each can simultaneously check and reinforce the other. It provides great protection from being overcome by blind, true-believing zeal and corrupting cynicism (which may be two sides of the same false coin). Shaw was a great teacher for many in my generation. I started reading him when I was in sixth grade, and I responded strongly not only to the wit but to various modes, scene and occasions of argument and debate as they were framed by various kinds of dramatic situation. I remember being electrified when quite young by the moment in the epilogue scene of Saint Joan when the English chaplain, De Stogumber, who had been so zealous in urging for Joan’s being burned at the stake, returns to testify about how seeing her suffering the flames had made a changed man of him. The Inquisitor, Peter Cauchon, calls out (with what I imagined was a kind of moral distaste I’d never been aware of before), "Must then a Christ perish in torment in every age to save those who have no imagination?" It introduced me to a skepticism about the self-satisfaction of the born-again, of any persuasion. With Auden and Orwell, much later on and after my mental world had become more complicated, it was education in negotiating a living way between a destructively naïve idealism and the crackpot realism—equally inimical to the pragmatic. PD: Would you consider yourself a "formal" pragmatist, i.e., a student of Peirce, James, Dewey, Mead (etc.) or an "informal" pragmatist – someone taking the common-sense position on events…or someone who refuses to be pigeon-holed politically? JH: "Informal" – of the sort that often leads me to ask of theoretical formulations, "Yes, but what’s it for ?" PD: Which other authors do you think might help us negotiate between "naïve idealism" and "crackpot realism"? I think of Joyce, Wallace Stevens, perhaps Faulkner? JH: When I was in college, a strong teacher for just this question was Cervantes. One feels, in an Emersonian way, that the Don’s view of the world is correct at midnight, and Sancho’s at noon. |
Then there is mathematical realism.
A post in this journal on Saturday, the reported date of Hollander's death,
discussed a possible 21st-century application of 19th-century geometry.
For some background, see Peter J. Cameron's May 11, 2010, remarks
on Sylvester's duads and synthemes . The following figure from the
paper discussed here Saturday is related to figures in Cameron's remarks.
New York Daily News , 2:55 PM EST today—
Joe Simon, who dreamed up the star-spangled super hero Captain America while riding on a Manhattan bus during the early days of World War II, died Thursday [Dec. 15] after an undisclosed illness. He was 98.
New York Times , about 10 PM EST today—
Joe Simon, a writer, editor and illustrator of comic books who was a co-creator of the superhero Captain America, conceived out of a patriotic impulse as war was roiling Europe, died on Wednesday [Dec. 14] at his home in Manhattan. He was 98.
The discrepancy is perhaps due to initial reports that quoted Simon's family as saying he died "Wednesday night."
Simon was a co-creator of Captain America. For some background on Simon and a photo with his fellow comic artist Jerry Robinson, co-creator of The Joker, see a Washington Post article from this afternoon. Robinson died on either Wednesday, Dec. 7, or Thursday, Dec. 8, 2011.
Jerry Robinson, a pioneer in the early days of Batman comics and a key force in the creation of Robin the Boy Wonder; the Joker; Bruce Wayne’s butler, Alfred; and Two-Face, died Wednesday afternoon [Dec. 7] in New York City. He was 89.
CNN—
Cartoonist Jerry Robinson, who worked on the earliest Batman comics and claimed credit for creating the super-villain The Joker, died Thursday [Dec. 8] at the age of 89, his family confirmed.
A picture by Robinson—
The Joker in January 1943
with a Nov. 27 calendar page
A non-joke from a more recent November 27—
Powered by WordPress
