"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 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.
Warren (PA) Public Library's Instagram
on January 21, 2022 —
Morphart —
Morph Art — from Raiders of the Lost Coordinates
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
"Leave a space." — Stoppard, "Jumpers."
"Read something that means something." — The New Yorker
Related material — Bialystock Configurations :
Malgorzata Prazmowska and Krzysztof Prazmowski,
University of Bialystok, Bialystok, Poland
malgpraz@math.uwb.edu.pl and krzypraz@math.uwb.edu.pl
A pyramid scheme† in memory of the late Bernie Madoff —
 |
The above passage from Whitehead’s 1906 book suggests
that the tetrahedral model may be older than Polster thinks.*
This is shown by . . .
† See also “Profzi Scheme.”
* For some related work of the above “D. Mesner,” see
Mesner, D. (1967). “Sets of Disjoint Lines in PG(3, q),”
Canadian Journal of Mathematics, 19, 273-280.
" LaTeX is widely used in academia[3][4]
for the communication and
publication of scientific documents
in many fields . . . ." — Wikipedia
Related academic remarks —
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."
Clay Risen memorializes an advertising executive —
“Ms. Young rose to prominence as an executive with Grey Advertising,
where she began in 1959, standing out as one of the few women and
one of the few Asian-Americans at the firm, which was then a power
in its field.”
Related art from Log24 on the date of Ms. Young’s death —
“A colour is eternal.
It haunts time like a spirit.”
— Alfred North Whitehead
“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
stand-in for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”
— Richards, Douglas E.. The Enigma Cube (Alien Artifact Book 1)
(pp. 160-161). Paragon Press, 2020. Kindle Edition.
The biblical apple also appears in the game, and film, Assassin's Creed .
Related material —
See the cartoon version of Alfred North Whitehead in the previous post,
and some Whitehead-related projective geometry —
"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eye-in-the-pyramid points.
there are, as it’ll turn out, 168 of ’em that’ll work."
— Comment at a weblog on November 27, 2010.
See also Log24 on that date.
The 11/27/2010 comment was on a post dated November 23, 2010.
See also Log24 on that date.
"Leonardo was something like what we now call a Conceptual artist,
maybe the original one. Ideas — experiments, theories — were
creative ends in themselves."
— Holland Cotter in the online New York TImes this evening
From other Log24 posts tagged Tetrahedron vs. Square —
* Phrase from the previous post, "Overarching Narrative."
(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) —
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
(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! —
Previous posts now tagged Pyramid Game suggest …
A possible New Yorker caption: " e . . . (ab) . . . (cd) . "
Caption Origins —
Playing with shapes related to some 1906 work of Whitehead:
|
Thursday, September 12, 2019
Tetrahedral Structures
|
Playing with shapes related to some 1906 work of Whitehead:
The disappearance of "Christo et Ecclesiae" at Harvard
Rabbit Hole
Memory Hole
The above Harvard seal in a PDF —
The same page, minus the seal, today at the Internet Archive —
For a larger image of the seal-less page, click here.
Click to enlarge.
From the former date above —
|
Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
Consider Stevens’s verse from “The Rock” (1954):
“That in which space itself is contained.”
Consider also Whitehead in 1906 —
"This is proved by the consideration
of a three dimensional geometry in which
there are only fifteen points."
— and Stevens on the sublime (1935):
"And the sublime comes down
To the spirit itself,
The spirit and space,
The empty spirit
In vacant space."
See also a Terry Gilliam film on "crunching entities."
From page 180, Logicomix —
Alfred North Whitehead in the first of
the above-named years, 1906 —
"But the project's central problem was always there."
"The deeper we got into our Quest…
…The more I doubted its premises."
— Attributed to Bertrand Russell
by Apostolos Doxiadis and Christos
Papadimitriou in Logicomix (2008-9)
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“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
Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)
The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).
|
|
The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.
Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square model,
based on the 4×4 array I call the Galois tesseract (at left below).
(Click to enlarge.)
Marshall McLuhan in "Annie Hall" —
"You know nothing of my work."
Related material —
"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"
— Paul Simon
It was a dark and stormy night…
— Page 180, Logicomix
A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—
See also Absolute Ambition (Nov. 19, 2010).
* For the title, see Vanishing Point in this journal.
Meanwhile…
|
Log24 on Sunday, October 5, 2008— “In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.” Related material: Terence McKenna: “Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey ‘haunts time like a ghost.’” Anonymous author: “‘Knight’s move thinking’ is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia.” |
Related journalism—
"What's the 'S' stand for?" — Amy Adams
For some background, see "Cartoon Graveyard" and "Many Dimensions."
In memory of Sir Andrew Huxley, OM, who died on May 30, 2012
C. P. Snow on G. H. Hardy at Trinity College, Cambridge—
He played his games and indulged his eccentricities.
He was living in some of the best intellectual company
in the world— G. E. Moore, Whitehead, Bertrand Russell,
Trevelyan, the high Trinity society which was shortly to
find its artistic complement in Bloomsbury. (Hardy himself
had links with Bloomsbury, both of personal friendship
and of sympathy.)
See also "Max Black" + Trinity in this journal.
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
Part I: True
Bulletin of the American Mathematical Society , October 2002, page 563 —
“… the study of symmetries of patterns led to… finite geometries….”
– David W. Henderson, Cornell University
This statement may be misleading, if not (see Part II below) actually false. In truth, finite geometries appear to have first arisen from Fano's research on axiom systems. See The Axioms of Projective Geometry by Alfred North Whitehead, Cambridge University Press, 1906, page 13.
Part II: Grid
For the story of how symmetries of patterns later did lead to finite geometries, see the diamond theorem.
New York Lottery yesterday, December 16, 2010— midday 282, evening 297.
Suggested by a Jesuit commentary that mentions the midday number —
Page 282 of Encyclopaedia of Religion and Ethics , Volume 2,
edited by James Hastings, John Alexander Selbie, and Louis Herbert Gray,
New York, Charles Scribner's Sons, 1910 —
"Philosophy seeks not absolute first principles, nor yet purely immediate insights,
but the self-mediation of the system of truth, and an insight into this self-mediation.
Axioms, in the language of modern theory, are best defined, neither as certainties
nor as absolutely first principles, but as those principles which are used as the first
in a special theory.
LITERATURE — A complete view of the literature of the problems
regarding axioms is impossible, since the topic is connected with all
the fundamental philosophical issues…. JOSIAH ROYCE"
Suggested by the evening number, 297, and by Amy Adams (see previous post) —
See also a cartoon version of Russell and Whitehead discussing axioms.
A reviewer says Steve Martin finds in his new novel An Object of Beauty "a sardonic morality tale."
From this journal on the day The Cube was published (see today's Art Object ) —
Monday February 20, 2006
|
See also a post on Mathematics and Narrative from Nov. 14, 2009.
That post compares characters in Many Dimensions to those in Logicomix—
On Misplaced Concreteness—
An excerpt from China and Vietnam: The Politics of Asymmetry, by Brantly Womack (Cambridge U. Press, 2006)—
The book is intended to be a contribution to the general theory of international relations as well as to the understanding of China and Vietnam, but I give greater priority to “the case” rather than to the theory. This is a deliberate methodological decision. As John Gerring has argued, case studies are especially appropriate when exploring new causal mechanisms.2 I would argue more broadly that the “case” is the reality to which the theory is secondary. In international relations theory, “realism” is often contrasted to “idealism,” but surely a more basic and appropriate meaning of “realism” is to give priority to reality rather than to theory. The philosopher Alfred North Whitehead defined the Fallacy of Misplaced Concreteness as “neglecting the degree of abstraction involved when an actual entity is considered merely so far as it exemplifies certain categories of thought.”3 In effect, the concept is taken as the concrete reality, and actual reality is reduced to a mere appendage of data. Misplaced Concreteness may well be the cardinal sin of modern social science. It is certainly pandemic in international relations theory, where a serious consideration of the complexities of real political situations is often dismissed as mere “area studies.” Like the Greek god Anteus who was sustained by touching his Mother Earth, theory is challenged and rejuvenated by planting its feet in thick reality.
2 John Gerring, "What Is a Case Study and What Is It Good For?"
American Political Science Review 98:2 (May 2004), pp. 341-54
3 Alfred North Whitehead, Process and Reality
(New York: Harper, 1929), p. 11
Remarks—
"Whitehead defined the Fallacy of Misplaced Concreteness…."
The phrase "misplaced concreteness" occurs in the title of a part of an exhibition, "Theme and Variations," by artist Josefine Lyche (Oslo, 2009). I do not know what Lyche had in mind when she used the phrase. A search for possible meanings yielded the above passage.
"In international relations theory, “realism” is often contrasted to “idealism….”
For a more poetic look at "realism" and "idealism" and international relations theory, see Midsummer Eve's Dream.
Battleship Gray
“A colour is eternal.
It haunts time like a spirit.
It comes and it goes.
But where it comes it is the same colour.
It neither survives nor does it live.
It appears when it is wanted.”
– Alfred North Whitehead,
Science and the Modern World, 1925
Another Opening of Another Show
"What we've got now isn't really healthcare reform, it's a reshuffling of the deck chairs on the Titanic as far as our patients are concerned, and we're going to make sure that we … have universal healthcare that is truly universal and has eliminated the insurance companies," she told Reuters.
— Deborah Burger, president-elect of the new 150,000-member National Nurses United, comprising union locals from Maine to Hawaii
A graphic novel reviewed in the current Washington Post features Alfred North Whitehead and Bertrand Russell–
Related material:
Whitehead on Fano's finite projective three-space:
"This is proved by the consideration of a three dimensional geometry in which there are only fifteen points."
—The Axioms of Projective Geometry , Cambridge University Press, 1906
Further reading:
See Solomon's Cube and the link at the end of today's previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams's portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .
in memory of
G. H. Hardy,
who died on
this date in 1947
"He was living in some of the best intellectual company in the world– G.E. Moore, Whitehead, Bertrand Russell, Trevelyan, the high Trinity society which was shortly to find its artistic complement in Bloomsbury."
For a rather different artistic complement, see the previous entry.
The previous two entries mention,
and illustrate, the color grey.
Another illustration, on the cover
of one of my favorite books:
"A colour is eternal.
It haunts time like a spirit."
— Alfred North Whitehead
From John Lahr's
winter 2002 review
of "Our Town"–
"We all know that something is eternal," the Stage Manager says. "And it ain't houses and it ain't names, and it ain't earth, and it ain't even stars– everybody knows in their bones that something is eternal, and that something has to do with human beings."
The Stage Manager was played by Paul Newman. The review was subtitled "Getting the Spirit Onstage."
Yesterday's entry contained the following unattributed quotation:
"One must join forces with friends of like mind."
As the link to Leap Day indicated, the source of the quotation is the I Ching.
Yesterday's entry also quoted the late Terence McKenna, a confused writer on psychosis and the I Ching. Lest the reader conclude that I consider McKenna or similar authors (for instance, Timothy Leary in Cuernavaca) as "friends of like mind," I would point rather to more sober students of the I Ching (cf. my June 2002 notes on philosophy, religion, and science) and to the late Scottish theologian John Macquarrie:
Macquarrie's connection in this journal to the I Ching is, like that book itself, purely coincidental. For details, click on the figure below.
The persistent reader will
find a further link that
leads to an entry titled
"Notes on the I Ching."
"A colour is eternal. It haunts time like a spirit. It comes and it goes. But where it comes it is the same colour. It neither survives nor does it live. It appears when it is wanted."
Last night's entry presented a
short story summarized by
four lottery numbers.
Today's mid-day lotteries
and associated material:
Pennsylvania, 201– i.e., 2/01:
Kindergarten Theology —
"In a game of chess, the knight's move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a 'knight's move' in intelligence."
"One must join forces with friends of like mind"
Related material:
"Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey 'haunts time like a ghost.'"
"'Knight's move thinking' is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia."
I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.
For more on the sleep of Apollo,
see the front page of today's
New York Times Book Review.
Garrison Keillor's piece there,
"Dying of the Light," is
about the fear of death felt
by an agnostic British twit.
For relevant remarks by
a British non-twit, see
William Dunbar–
|
"Logos and logic, crystal hypothesis,
— Wallace Stevens, |
Yesterday's meditation ("Simon's Shema") on the interpenetration of opposites continues:
"The fundamental conception of Tantric Buddhist metaphysics, namely, yuganaddha, signifies the coincidence of opposites. It is symbolized by the conjugal embrace (maithuna or kama-kala) of a god and goddess or a Buddha and his consort (signifying karuna and sunyata or upaya and prajna, respectively), also commonly depicted in Tantric Buddhist iconography as the union of vajra (diamond sceptre) and padme (lotus flower). Thus, yuganaddha essentially means the interpenetration of opposites or dipolar fusion, and is a fundamental restatement of Hua-yen theoretic structures."
— p. 148 in "Part II: A Whiteheadian Process Critique of Hua-yen Buddhism," in Process Metaphysics and Hua-Yen Buddhism: A Critical Study of Cumulative Penetration vs. Interpenetration (SUNY Series in Systematic Philosophy), by Steve Odin, State University of New York Press, 1982
And on p. 163 of Odin, op. cit., in "Part III: Theology of the Deep Unconscious: A Reconstruction of Process Theology," in the section titled "Whitehead's Dipolar God as the Collective Unconscious"–
"An effort is made to transpose Whitehead's theory of the dipolar God into the terms of the collective unconscious, so that now the dipolar God is to be comprehended not as a transcendent deity, but the deepest dimension and highest potentiality of one's own psyche."
Odin obtained his Ph.D. degree from the Department of Philosophy at the State University of New York (SUNY) at Stony Brook in 1980. (See curriculum vitae (pdf).)
For an academic review of Odin's book, see David Applebaum, Philosophy East and West, Vol. 34 (1984), pp. 107-108.
It is perhaps worth noting, in light of the final footnote of Mark D. Brimblecombe's Ph.D. thesis "Dipolarity and God" quoted yesterday, that "tantra" is said to mean "loom." For some less-academic background on the Tantric iconography Odin describes, see the webpage "Love and Passion in Tantric Buddhist Art." For a fiction combining love and passion with the word "loom" in a religious context, see Clive Barker's Weaveworld. This fiction– which is, if not "supreme" in the Wallace Stevens sense, at least entertaining– may correspond to some aspects of the deep Jungian psychological reality discussed by Odin.
Click on image for details.
Haunting Time
"Macquarrie remains one of the most
important commentators [on] …
Heidegger's work. His co-translation
of Being and Time into English is
considered the canonical version."
— Wikipedia
The Rev. Macquarrie died on
May 28. The Log24 entry
for that date contains the
following illustration:
The part of the illustration
relevant to the death of
Macquarrie is the color.
From my reply to
a comment on the
May 28 entry:
From McKenna's afterword:
My own remarks on the hippie
scene seem appropriate as a
response to media celebration
of today's 40th anniversary of
the beginning of the 1967
"summer of love."
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