Call a 4×4 array labeled with 4 copies each
of 4 different symbols a foursquare.
The symmetries of foursquares are governed
by the symmetries of their 24 interstices —
(Cullinane, Diamond Theory, 1976.)
From Log24 posts tagged Mathieu Cube —
A similar exercise might involve the above 24 interstices of a 4×4 array.
Perhaps Crossan should have consulted Galois, not Piaget . . .
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
Epigraph to The Dark Interval , by John Dominic Crossan —
“I am the pause between two notes that fall
into a real accordance scarce at all:
for Death’s note tends to dominate—
Both, though, are reconciled in the dark interval,
tremblingly.
And the song remains immaculate.”
—Rilke, The Book of Hours , I
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy in A Mathematician's Apology
For Bill Irwin —
<meta property="article:published_time"
content="2023-02-06T11:00:00.000Z"/>
The Source —
view-source:https://www.newyorker.com/magazine/2023/02/13/
finding-laughs-amid-the-gray-in-becketts-endgame
See as well the previous post and . . .
Patterns in the space
between subsquares —
Figures in the space
between fingers —
From Wednesday, St. Bridget's Day, 2023 —
Poetic meditation from The New Yorker today —
"If the tendency of rhyme, like that of desire,
is to pull distant things together
and force their boundaries to blur,
then the countervailing force in this book,
the one that makes it go, is the impulse
toward narrative, toward making sense of
the passage of time."
Last updated at 22:46 PM ET on 1 February 2023.
Click for a designer's obituary.
Paraphrase for a road-sign collector:
See as well … Today's New York Times obituary
of the Harvard Business School Publishing
Director of Intellectual Property.
From a search in this journal for Godard —
"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."
— Jean-Luc Godard, "Introduction à une véritable histoire
du cinéma," Albatros , Paris, 1980, p. 145
Architectural theorist Jeffrey Kipnis in 1991, recalled here in 2015 —
For the source of the illustration, see Hexagram 14.
"Godard, in the final analysis, expands the Warburgian programme
of iconology into that of a cinematographic iconology of the interstice."
— The author of the essay quoted in the previous post.
From Martin Heidegger's
Contributions to Philosophy (From Enowning) ,
Translated by Parvis Emad and Kenneth Maly,
Indiana University Press, 1999 (first published in German
in 1989 as Beiträge zur Philosophie (Vom Ereignis) but
written in 1936-1938 —
"The 'between' [das Zwischen ] is the simple 'bursting open'
that enowns be-ing to a being, which up until then is held back
from what is ownmost to it and is not yet to be named a being.
This 'bursting open' is the clearing for the sheltered. But the
'bursting open' does not disperse. and the clearing is not a mere
emptiness.
The 'between' [das Zwischen ] which bursts open gathers
what it removes into the open of its strifing and refusing
belongingness, moves unto the ab-ground , out of which everything
(god, man, world, earth) recoils in swaying into itself and thus leaves
to be-ing the unique decidedness of en-ownment."
— 270, "The Essential Sway of Be-ing" (p. 341)
"Enownment and enstrifing, historical grounding and decision,
uniqueness and the onefold, what has the character of
the between [Zwischenhafte ] and the cleavage [Geklüft ] —
they never name the essential sway of be-ing as properties
but rather in each case the whole essential swaying* of its essential
sway."
— 270, "The Essential Sway of Be-ing" (p. 342)
* For "swaying" as "unfolding," see (for instance)
the Stanford Encyclopedia of Philosophy and
also George Kovacs, Thinking and Be-ing in Heidegger's
Beiträge zur Philosophie (Vom Ereignis) ,
Zeta Books, 2015.
"… nothing could be demarcated as 'hors d'oeuvre'…"
— Geoffrey Hartman in his Haskins Lecture for 2000
(quoted here on Columbus Day, 2004).
See also May Day 2016 and Gap Dance.
From Wallace Stevens, "The Man with the Blue Guitar":
IX
And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .
"Arrowy, still strings" from the diamond theorem
"Honored in the Breach:
Graham Bader on Absence as Memorial"
Artforum International , April 2012
|
. . . . "In the wake of a century marked by inconceivable atrocity, the use of emptiness as a commemorative trope has arguably become a standard tactic, a default style of public memory. The power of the voids at and around Ground Zero is generated by their origin in real historical circumstance rather than such purely commemorative intent: They are indices as well as icons of the losses they mark.
Nowhere is the negotiation between these two possibilities–on the one hand, the co-optation of absence as tasteful mnemonic trope; on the other, absence's disruptive potential as brute historical scar–more evident than in Berlin, a city whose history, as Andreas Huyssen has argued, can be seen as a 'narrative of voids.' Writing in 1997, Huyssen saw this tale culminating in Berlin's post-wall development, defined equally by an obsessive covering-over of the city's lacunae–above all in the elaborate commercial projects then proliferating in the miles-long stretch occupied until 1989 by the Berlin Wall–and a carefully orchestrated deployment of absence as memorial device, particularly in the 'voids' integrated by architect Daniel Libeskind into his addition to the Berlin Museum, now known as the Jewish Museum Berlin." |
See also Espacement and The Thing and I.
Correction — "Death has 'the whole spirit sparkling…'"
should be "Peace after death has 'the whole spirit sparkling….'"
The page number, 373, is a reference to Wallace Stevens:
Collected Poetry and Prose , Library of America, 1997.
See also the previous post, "Critical Invisibility."
From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 59-79 (1992):
"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."
On "The Emperor's New Clothes" —
|
Andersen’s weavers, as one commentator points out, are merely insisting that “the value of their labor be recognized apart from its material embodiment.” The invisible cloth they weave may never manifest itself in material terms, but the description of its beauty (“as light as spiderwebs” and “exquisite”) turns it into one of the many wondrous objects found in Andersen’s fairy tales. It is that cloth that captivates us, making us do the imaginative work of seeing something beautiful even when it has no material reality. Deeply resonant with meaning and of rare aesthetic beauty—even if they never become real—the cloth and other wondrous objets d’art have attained a certain degree of critical invisibility. — Maria Tatar, The Annotated Hans Christian Andersen (W. W. Norton & Company, 2007). Kindle Edition. |
(Continued from the previous post.)
|
In-Between "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' 1), writing (of which 'spacing' is said to be 'the fundamental property' 2) and deconstruction (with one of Derrida's last major texts, Le Toucher: Jean-Luc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' 3)." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, Jean-Luc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,16 we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207-227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'in-between' more 'originary' than the play of differences it in-forms. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a pre-ontological 'opening' – or, shall we say, 'in-between.' |
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
|
https://www.parrhesiajournal.org/ Parrhesia No. 3 • 2007 • 22–32
(Up) Against the (In) Between: Interstitial Spatiality by Clare Blackburne Blackburne — www.parrhesiajournal.org 24 — "The excessive notion of espacement as the resurgent spatiality of that which is supposedly ‘without space’ (most notably, writing), alerts us to the highly dynamic nature of the interstice – a movement whose discontinuous and ‘aberrant’ nature requires further analysis." Blackburne — www.parrhesiajournal.org 25 — "Espacement also evokes the ambiguous figure of the interstice, and is related to the equally complex derridean notions of chora , différance , the trace and the supplement. Derrida’s reading of the Platonic chora in Chora L Works (a series of discussions with the architect Peter Eisenman) as something which defies the logics of non-contradiction and binarity, implies the internal heterogeneity and instability of all structures, neither ‘sensible’ nor ‘intelligible’ but a third genus which escapes conceptual capture.25 Crucially, chora , spacing, dissemination and différance are highly dynamic concepts, involving hybridity, an ongoing ‘corruption’ of categories, and a ‘bastard reasoning.’26 Derrida identification of différance in Margins of Philosophy , as an ‘unappropriable excess’ that operates through spacing as ‘the becoming-space of time or the becoming-time of space,’27 chimes with his description of chora as an ‘unidentifiable excess’ that is ‘the spacing which is the condition for everything to take place,’ opening up the interval as the plurivocity of writing in defiance of ‘origin’ and ‘essence.’28 In this unfolding of différance , spacing ‘insinuates into presence an interval,’29 again alerting us to the crucial role of the interstice in deconstruction, and, as Derrida observes in Positions , its impact as ‘a movement, a displacement that indicates an irreducible alterity’: ‘Spacing is the impossibility for an identity to be closed on itself, on the inside of its proper interiority, or on its coincidence with itself. The irreducibility of spacing is the irreducibility of the other.’30"
25. Quoted in Jeffrey Kipnis and Thomas Leeser, eds., 26. Ibid, 25.
27. Derrida, Margins of Philosophy. 28. Derrida, Chora L Works , 19 and 10. 29. Ibid, 203. 30. Derrida, Positions , 94. |
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
Finite Galois geometry with the underlying field the simplest one possible —
namely, the two-element field GF(2) — is a geometry of interstices :
For some less precise remarks, see the tags Interstice and Interality.
The rationalist motto "sincerity, order, logic and clarity" was quoted
by Charles Jencks in the previous post.
This post was suggested by some remarks from Queensland that
seem to exemplify these qualities —
For PSL(2,7), this is ((49-1)(49-7))/((7-1)(2))=168.
The group GL(3,2), also of order 168, acts naturally
on the set of seven cube-slicings below —
Another way to picture the seven natural slicings —
Application of the above images to picturing the
isomorphism of PSL(2,7) with GL(3,2) —
.gif)
For a more detailed proof, see . . .
The title is from Warburg. The Zwischenraum lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
set-theoretic symmetric difference operator.
|
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by Gian-Carlo Rota Some results from a Google search — Eidetic reduction | philosophy | Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. |
For example —
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.
In this field, "1 + 1 = 0," as noted here Wednesday.
See also other posts now tagged Interstice.
The title is from the 2013 paper by Latsis in the previous post.
The symmetries of the interstices at right underlie
the symmetries of the images at left.
The previous post, "One Plus One," suggests some further
art-historical remarks on interality —
|
From Third Text , 2013, Vol. 27, No. 6, pp. 774–785 — "Genealogy of the Image in Histoire(s) du Cinéma : Godard, Warburg and the Iconology of the Interstice" * * * * P. 775 — My discussion will focus on the significance of the concept of the ‘space in-between,’ its importance for Godard’s work and its role in a relational historiography of images more broadly. I hope to corroborate how Godard functions as a twenty-first century archaeologist of the moving image, constructing a meta-cinematic collage that, while consisting of an indexing of (almost exclusively) pre-existing filmic samples, ends up becoming a hybrid work of art in its own right. Godard, in the final analysis, expands the Warburgian programme of iconology into that of a cinematographic iconology of the interstice. * * * * P. 777 — Godard conceives of the image only in the plural, in the intermediate space between two images, be it a prolonged one (in Histoire(s) there are frequent instances of black screens) or a non-existent one (superimposition, co-presence of two images on screen). He comments: ‘[For me] it’s always two, begin by showing two images rather than one, that’s what I call image, the one made up of two’ [18] and elsewhere, ‘I perceived . . . cinema is that which is between things, not things [themselves] but between one and another.’ [19] 18. Jean-Luc Godard and Youssef Ishaghpour, "Archéologie du cinéma et mémoire du siècle," Farrago ,Tours, 2000, p. 27. The title of this work is reflective of the Godardian agenda that permeates Histoire(s) . 19. Jean-Luc Godard, "Introduction à une véritable histoire du cinéma," Albatros , Paris,1980, p. 145 * * * * P. 783 — If it is in ‘the in-between’ that thought is born, then for Godard cinematography as ‘a form that thinks . . . was born with the advent of modern painting.’ [62] 62. Godard and Ishaghpour, op. cit., pp 45–46. * * * * P. 785 — Warburg commented on the signification of the black spaces that he placed between images in his analysis of the network of intervals in Mnemosyne , by quoting Johann Wolfgang Goethe’s dictum ‘the truth inhabits the middle space.’ [68] This citation induces a feeling of déjà-vu for the viewer of Histoire(s). The link was not missed by Warburg himself, as one of his diary entries testifies: ‘We can compare this phenomenon [the iconology of the interval] to that of the cinematic montage, the domain of the interpretation is an intervallic one.’ [69] 68. Warburg, Mnemosyne , pp 135–146. 69. Warburg is quoted in Didi-Huberman, L’image survivante, p. 503. (Georges Didi-Huberman, L’image survivante. Histoire de l’art et temps des fantômes selon Aby Warburg , Minuit, Paris, 2002) |
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