Tuesday, August 13, 2019

Putting the Structure  in Structuralism

Filed under: General — Tags: , , — m759 @ 8:34 PM

The Matrix of Lévi-Strauss —

(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée 
, No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)

The structure  of the matrix of Lévi-Strauss —

Illustration from Diamond Theory , by Steven H. Cullinane (1976).

The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.

Thursday, August 6, 2020

Structure and Mutability . . .

Filed under: General — Tags: , — m759 @ 8:33 PM


See a Log24 search for Beadgame Space.

This  post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .

Related material from a Log24 search for Structuralism

IMAGE- Stella Octangula and Claude Levi-Strauss

Saturday, August 17, 2019

Crystalline Complexity

Filed under: General — Tags: — m759 @ 6:08 PM

Burroway on Hustvedt in The New York Times ,
Sunday, March 9, 2003 —

See as well "Putting the Structure  in Structuralism."

Saturday, November 5, 2016


Filed under: General — m759 @ 7:31 PM

Wikipedia— "The first Million Mask March occurred in 2013."

A check of the date of that march in this journal yields

See as well, more generally, "Interpenetration" in this journal.

Thursday, November 12, 2015

The Raw, the Cooked, and the Spoiled

Filed under: General — Tags: — m759 @ 12:00 PM

On the late French philosopher André Glucksmann,
a paragraph from The New Yorker

The style could be overwhelming at times, and was
often a more effective instrument of intellectual
pleasure than political persuasion. But, in return,
it produced a thousand small epiphanies—
for instance, his lovely mordant point, made at length
in one of his books, that between the “raw” and the
“cooked”—the simple binary beloved of structuralism
there was always the “pourri,” the rotting, the rotten.
Our refusal to take in the rotting as a category of its own
was, he suggested, with a delighted literary grimace,
a kind of moral blindness, part of a fake dialectic that
blinded us to the muddled, rotting truth of the world.
The real world was not composed of oscillating
dialectical forces; it was composed of actual suffering
people crushed between those forces. — Adam Gopnik

See also

Click the above image for some backstory.

Thursday, November 7, 2013

Pattern Grammar

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

Yesterday afternoon's post linked to efforts by
the late Robert de Marrais to defend a mathematical  
approach to structuralism and kaleidoscopic patterns. 

Two examples of non-mathematical discourse on
such patterns:

1.  A Royal Society paper from 2012—

Click the above image for related material in this journal.

2.  A book by Junichi Toyota from 2009—

Kaleidoscopic Grammar: Investigation into the Nature of Binarism

I find such non-mathematical approaches much less interesting
than those based on the mathematics of reflection groups . 

De Marrais described the approaches of Vladimir Arnold and,
earlier, of H. S. M. Coxeter, to such groups. These approaches
dealt only with groups of reflections in Euclidean  spaces.
My own interest is in groups of reflections in Galois  spaces.
See, for instance, A Simple Reflection Group of Order 168

Galois spaces over fields of characteristic 2  are particularly
relevant to what Toyota calls binarism .

Wednesday, November 6, 2013

Bullshit Studies

Filed under: General — Tags: — m759 @ 3:12 PM

The essay excerpted in last night's post on structuralism
is of value as part of a sustained attack by the late
Robert de Marrais on the damned nonsense of the late
French literary theorist Jacques Derrida—

Catastrophes, Kaleidoscopes, String Quartets:
Deploying the Glass Bead Game

Part I:  Ministrations Concerning Silliness, or:
Is “Interdisciplinary Thought” an Oxymoron?

Part II:  Canonical Collage-oscopes, or:
Claude in Jacques’ Trap?  Not What It Sounds Like!

Part III:  Grooving on the Sly with Klein Groups

Part IV:  Claude’s Kaleidoscope . . . and Carl’s

Part V:  Spelling the Tree, from Aleph to Tav
(While  Not Forgetting to Shin)

The response of de Marrais to Derrida's oeuvre  nicely
exemplifies the maxim of Norman Mailer that

"At times, bullshit can only be countered
with superior bullshit."

Tuesday, November 5, 2013

For St. Robert de Marrais

Filed under: General — m759 @ 11:01 PM

Bead-Game Structuralism:
Excerpts from Comments by Robert de Marrais
on Interpenetration and The Raw and the Cooked

Click the image below for the webpage:

Tuesday, November 29, 2011

The Flight from Ennui

Filed under: General — Tags: — m759 @ 12:00 PM

Post 2310 in yesterday evening’s Short Story links to two posts
from 2006 inspired by Oxford mathematician Marcus du Sautoy—

Thursday, May 25, 2006


May there be an ennui
of the first idea?
What else, prodigious scholar,
should there be?

— Wallace Stevens,
“Notes Toward a Supreme Fiction”

Related material: The Line.

7:13 PM

Order and Ennui

Meanwhile, back at the Institute
for Advanced Study:

May 25, 4:40 PM —
Research Seminar
(Simonyi Hall Seminar Room) —
Pirita Paajanen,
The Hebrew University of Jerusalem:
Zeta functions of
finitely generated infinite groups

Some background cited by Paajanen:

M.P.F. du Sautoy,
“Zeta functions of groups:
The quest for order
versus the flight from ennui,”
Groups St Andrews 2001 in Oxford ,
Volume 1, CUP 2003.

Those who prefer the showbiz
approach to mathematics
(the flight from ennui?) may
enjoy a website giving
further background from du Sautoy.

4:40 PM

The first paragraph of
Zeta Functions of Groups: The Quest for Order
Versus the Flight from Ennui
,” by Marcus du Sautoy,
Mathematical Institute, University of Oxford—

“Mathematics is about the search for patterns,
to see order where others see chaos. We are very lucky
to find ourselves studying a subject which is neither so rigid
that the patterns are easy, yet not too complicated
lest our brains fail to master its complexities.
John Cawelti sums up this interplay perfectly in a book*
not about mathematics but about mystery and romance:
‘if we seek order and security, the result is likely to be
boredom and sameness. But rejecting order for the sake
of change and novelty brings danger and uncertainty…
the history of culture can be interpreted as a dynamic
tension between these two basic impulses…
between the quest for order and the flight from ennui.”’

* John G. Cawelti, Adventure, Mystery, and Romance:
Formula Stories as Art and Popular Culture 
University of Chicago Press, 1976.

[Cawelti cites as his souce on interpreting “the history
of culture” Harry Berger, Jr., “Naive Consciousness and
Culture Change: An Essay in Historical Structuralism
Bulletin of the Midwest Modern Language Association ,
Vol. 6, No. 1 (Spring 1973): page 35.]

Here du Sautoy paints mathematicians as seekers of order,
apparently not realizing that the author he approvingly quotes
states that seekers of order face the danger of boredom.

Another danger to seekers
of order is, of course, seeing
order where there is none.


Are you the butterfly?

Sunday, December 19, 2010


Filed under: General,Geometry — Tags: , — m759 @ 8:00 AM

In the spirit of last night's SNL Hanukkah version of "It's a Wonderful Life" —

A Geometric Merkabah for Hanukkah from December 1, and…

"The Homepage of Contemporary Structuralism" (click to enlarge) —


Detail —


Summary —

IMAGE- Stella Octangula and Claude Levi-Strauss

Wednesday, November 4, 2009

Sinner or Saint?

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

As noted here yesterday, Claude Levi-Strauss may have died on Devil's Night, on Halloween, or on All Saints' Day. He was apparently a myth-transformer to the end.

The Independent says today he died on Sunday, All Saints' Day. Its eulogy, by Adam Kuper, is well-written, noting that linguist Roman Jakobson was a source of Levi-Strauss's theory of oppositions in myth, and observing that

"… binary oppositions tend to accumulate to form structures…."

Yes, they do. Examples:

I. The structures in the Diamond Puzzle

Adam and God (Sistine Chapel), with Jungian Self-Symbol and Ojo de Dios (The Diamond Puzzle)

Click on image for Jungian background.

II: The structure on a recent cover of Semiotica


Click to enlarge.

The Semiotica article by mathematical linguist Solomon Marcus is a defense of the Levi-Strauss canonic formula mentioned here yesterday.

It is available online for $40.

A less expensive, and possibly more informative, look at oppositions in linguistics is available for free online in a 1984 master's thesis (pdf, 8+ mb)–

"Language, Linguistics, and Philosophy: A Comparison of the Work of Roman Jakobson and the Later Wittgenstein, with Some Attention to the Philosophy of Charles Saunders Peirce," by Miles Spencer Kimball.

Tuesday, April 29, 2008

Tuesday April 29, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 11:09 AM
Sacerdotal Jargon
at Harvard:

Thomas Wolfe

Thomas Wolfe
(Harvard M.A., 1922)


Rosalind Krauss

Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)


The Kernel of Eternity

"No culture has a pact with eternity."
George Steiner, interview in  
The Guardian of April 19

"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."

— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005


From today's online Harvard Crimson:

"… under the leadership of Faust,
Harvard students should look forward
to an ever-growing opportunity for
international experience
and artistic endeavor."


Wolfgang Pauli as Mephistopheles

Pauli as Mephistopheles
in a 1932 parody of
Faust at Niels Bohr's
institute in Copenhagen

From a recent book
on Wolfgang Pauli,
The Innermost Kernel:

Pauli's Dream Square (square plus the two diagonals)

A belated happy birthday
to the late
Felix Christian Klein
  (born on April 25) —

The Klein Group: The four elements in four colors, with black points representing the identity

Another Harvard figure quoted here on Dec. 5, 2002:

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."

— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)

From a review of Rosalind Krauss's The Optical Unconscious  (MIT Press hardcover, 1993):

Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid-19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."

— Arthur C. Danto in ArtForum, Summer 1993

Rosalind Kraus in The Optical Unconscious (MIT Press paperback, 1994):

For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude Lévi-Strauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.

For related non-sacerdotal jargon, see…

Wikipedia on the Klein group (denoted V, for Vierergruppe):

In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.

For radicals of another sort, see A Logocentric Meditation, A Mass for Lucero, and [update of 7 PM] Steven Erlanger in today's New York Times— "France Still Divided Over Lessons of 1968 Unrest."

For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:

April 29, 2008, Google search on 'penrose space time'

Click on the above
 image for details.

See also yesterday's
Religious Art.

Monday, January 26, 2004

Monday January 26, 2004

Filed under: General — Tags: — m759 @ 1:11 PM

Language Game

More on "selving," a word coined by the Jesuit poet Gerard Manley Hopkins.  (See Saturday's Taking Lucifer Seriously.)

"… through the calibrated truths of temporal discipline such as timetabling, serialization, and the imposition of clock-time, the subject is accorded a moment to speak in."

Dr. Sally R. Munt,

Intelligibility, Identity, and Selfhood:
A Reconsideration of
Spatio-Temporal Models

The "moment to speak in" of today's previous entry, 11:29 AM, is a reference to the date 11/29 of last year's entry

Command at Mount Sinai.

That entry contains, in turn, a reference to the journal Subaltern Studies.  According to a review of Reading Subaltern Studies,

"… the Subaltern Studies collective drew upon the Althusser who questioned the primacy of the subject…."

Munt also has something to say on "the primacy of the subject" —

"Poststructuralism, following particularly Michel Foucault, Jacques Derrida and Jacques Lacan, has ensured that 'the subject' is a cardinal category of contemporary thought; in any number of disciplines, it is one of the first concepts we teach to our undergraduates. But are we best served by continuing to insist on the intellectual primacy of the 'subject,' formulated as it has been within the negative paradigm of subjectivity as subjection?"

How about objectivity as objection?

I, for one, object strongly to "the Althusser who questioned the primacy of the subject."

This Althusser, a French Marxist philosopher by whom the late Michael Sprinker (Taking Lucifer Seriously) was strongly influenced, murdered his wife in 1980 and died ten years later in a lunatic asylum.

For details, see

The Future Lasts a Long Time.


For details of Althusser's philosophy, see the oeuvre of Michael Sprinker.

For another notable French tribute to Marxism, click on the picture at left.

Friday, July 25, 2003

Friday July 25, 2003

Filed under: General — Tags: — m759 @ 5:24 PM

For Jung’s 7/26 Birthday:
A Logocentric Meditation

Leftist academics are trying to pull a fast one again.  An essay in the most prominent American mathematical publication tries to disguise a leftist attack on Christian theology as harmless philosophical woolgathering.

In a review of Vladimir Tasic’s Mathematics and the Roots of Postmodern Thought, the reviewer, Michael Harris, is being less than candid when he discusses Derrida’s use of “logocentrism”:

“Derrida uses the term ‘logocentrism’… as ‘the metaphysics of phonetic writing’….”

Notices of the American Mathematical Society, August 2003, page 792

We find a rather different version of logocentrism in Tasic’s own Sept. 24, 2001, lecture “Poststructuralism and Deconstruction: A Mathematical History,” which is “an abridged version of some arguments” in Tasic’s book on mathematics and postmodernism:

“Derrida apparently also employs certain ideas of formalist mathematics in his critique of idealist metaphysics: for example, he is on record saying that ‘the effective progress of mathematical notation goes along with the deconstruction of metaphysics.’

Derrida’s position is rather subtle. I think it can be interpreted as a valiant sublation of two completely opposed schools in mathematical philosophy. For this reason it is not possible to reduce it to a readily available philosophy of mathematics. One could perhaps say that Derrida continues and critically reworks Heidegger’s attempt to ‘deconstruct’ traditional metaphysics, and that his method is more ‘mathematical’ than Heidegger’s because he has at his disposal the entire pseudo-mathematical tradition of structuralist thought. He has himself implied in an interview given to Julia Kristeva that mathematics could be used to challenge ‘logocentric theology,’ and hence it does not seem unreasonable to try looking for the mathematical roots of his philosophy.”

The unsuspecting reader would not know from Harris’s review that Derrida’s main concern is not mathematics, but theology.  His ‘deconstruction of metaphysics’ is actually an attack on Christian theology.

From “Derrida and Deconstruction,” by David Arneson, a University of Manitoba professor and writer on literary theory:

Logocentrism: ‘In the beginning was the word.’ Logocentrism is the belief that knowledge is rooted in a primeval language (now lost) given by God to humans. God (or some other transcendental signifier: the Idea, the Great Spirit, the Self, etc.) acts a foundation for all our thought, language and action. He is the truth whose manifestation is the world.”

Some further background, putting my July 23 entry on Lévi-Strauss and structuralism in the proper context:

Part I.  The Roots of Structuralism

“Literary science had to have a firm theoretical basis…”

Part II.  Structuralism/Poststructuralism

“Most [structuralists] insist, as Levi-Strauss does, that structures are universal, therefore timeless.”

Part III.  Structuralism and
Jung’s Archetypes

Jung’s “theories, like those of Cassirer and Lévi-Strauss, command for myth a central cultural position, unassailable by reductive intellectual methods or procedures.”

And so we are back to logocentrism, with the Logos — God in the form of story, myth, or archetype — in the “central cultural position.”

What does all this have to do with mathematics?  See

Plato’s Diamond,

Rosalind Krauss on Art –

“the Klein group (much beloved of Structuralists)”

Another Michael Harris Essay, Note 47 –

“From Krauss’s article I learned that the Klein group is also called the Piaget group.”

and Jung on Quaternity:
Beyond the Fringe –

“…there is no denying the fact that [analytical] psychology, like an illegitimate child of the spirit, leads an esoteric, special existence beyond the fringe of what is generally acknowledged to be the academic world.”

What attitude should mathematicians have towards all this?

Towards postmodern French
atheist literary/art theorists –

Mathematicians should adopt the attitude toward “the demimonde of chic academic theorizing” expressed in Roger Kimball’s essay, Feeling Sorry for Rosalind Krauss.

Towards logocentric German
Christian literary/art theorists –

Mathematicians should, of course, adopt a posture of humble respect, tugging their forelocks and admitting their ignorance of Christian theology.  They should then, if sincere in their desire to honestly learn something about logocentric philosophy, begin by consulting the website

The Quest for the Fiction of an Absolute.

For a better known, if similarly disrespected, “illegitimate child of the spirit,” see my July 22 entry.

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