A transcription—
"Now suppose that α is an element of order 23 in M 24 ; we number the points of Ω
as the projective line ∞, 0, 1, 2, … , 22 so that
fact there is a full L 2 (23) acting on this line and preserving the octads…."
— R. T. Curtis, "A New Combinatorial Approach to M 24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society (1976),
* The title is of course a reference to the Knoxville of the previous post.
For art more closely related to the title "Alpha and Omega,"
see a different view of the above Hoyersten exhibition.
Wallace Stevens —
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
The title pun is from a letter about Google's new parent
company, named "Alphabet" by its founders.
Related material: Alpha and Omega in this journal.
"Looking for what was, where it used to be" — Wallace Stevens
— "Is this your business?"
— "No, but this is."
For Guy Fawkes Day, images from first and last posts —
an alpha and an omega of sorts —
from this journal in the month of December 2021 . . .
Some remarks on an artist who reportedly died
on the second day of that month —
"Even though he’s not currently part of a Star Wars movie,
Lindelof seems open to working on the franchise down the line.
'Will I get back in line outside the club and try to get back in again?
Of course,' he added. '[Star Wars ] was the alpha and the omega.
It’s the first movie I saw in a movie theater. I love all of the storytelling
in that world. If at first you don’t succeed, try, try again. Or again, again,
try, as Yoda would say.' ” — Jay Peters, The Verge , April 28, 2023
Α, ϴ, Ω
Related line:
Also from a Culture Desk of sorts:
Related art — Background colors for the letters in the NPR logo —
From Wallace Stevens —
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
From The Point magazine yesterday, October 8, 2019 —
Parricide: On Irad Kimhi's Thinking and Being .
Book review by Steven Methven.
The conclusion:
"Parricide is nothing that the philosopher need fear . . . .
What sustains can be no threat. Perhaps what the
unique genesis of this extraordinary work suggests is that
the true threat to philosophy is infanticide."
This remark suggests revisiting a post from Monday —
|
Monday, October 7, 2019
Berlekamp Garden vs. Kinder Garten
|
Stevens's Omega and Alpha (see previous post) suggest a review.
Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.
Illustrations —
The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
Or: Je repars .
From Wallace Stevens —
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
Mathematician Hanfried Lenz reportedly died in Berlin on June 1, 2013.
This journal that weekend —
A post suggested by remarks of J. D. Salinger in
The New Yorker of November 19, 1955 —
|
Wikipedia: Taiji (philosophy) Etymology The word 太極 comes from I Ching : "易有太極，是生兩儀，兩儀生四象，四象生八卦，八卦定吉凶，吉凶生大業。" Taiji (太極) is a compound of tai 太 "great; grand; supreme; extreme; very; too" (a superlative variant of da 大 "big; large; great; very") and ji 極 "pole; roof ridge; highest/utmost point; extreme; earth's pole; reach the end; attain; exhaust". In analogy with the figurative meanings of English pole, Chinese ji 極 "ridgepole" can mean "geographical pole; direction" (e.g., siji 四極 "four corners of the earth; world's end"), "magnetic pole" (Beiji 北極 "North Pole" or yinji 陰極 "negative pole; cathode"), or "celestial pole" (baji 八極 "farthest points of the universe; remotest place"). Combining the two words, 太極 means "the source, the beginning of the world". Common English translations of the cosmological Taiji are the "Supreme Ultimate" (Le Blanc 1985, Zhang and Ryden 2002) or "Great Ultimate" (Chen 1989, Robinet 2008); but other versions are the "Supreme Pole" (Needham and Ronan 1978), "Great Absolute", or "Supreme Polarity" (Adler 1999). |
See also Polarity in this journal.
* A phrase adapted, via Salinger,
from a poem by Sappho—
Ἴψοι δὴ τὸ μέλαθρον,
Υ᾽μήναον
ἀέρρετε τέκτονεσ ἄνδρεσ,
Υ᾽μήναον
γάμβροσ ἔρχεται ἶσοσ Ά᾽ρευϊ,
[Υ᾽μήναον]
ανδροσ μεγάλο πόλυ μείζων
[Υ᾽μήναον]
Πολυμερῶς καὶ πολυτρόπως πάλαι ὁ Θεὸς λαλήσας . . . .
( Long Day's Journey into Nighttown continues. )
"First and last, he was a skeptic …."
— Home page of Martin-Gardner.org
See also, in this journal, Alpha and Omega.
Related material from the last full day of Gardner's life —
See as well Symplectic in this journal.
The previous post dealt with one of the 64 symbols
(in a redesigned format) of the ancient Chinese classic
The I Ching .
For those who prefer to be guided by programmed
responses to alphabetical symbols …
A lyric by Ira Gershwin —
A cinematic "T" —
See also "T for Texas" in this journal and
George Clooney's recent attempt to commercialize
both the space program and the letter Omega:
From a post of May 13, 2015 —
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
From the series of posts tagged "Defining Form" —
The 4-point affine plane A and
the 7-point projective plane PA —
The circle-in-triangle of Yale's Figure 30b (PA ) may,
if one likes, be seen as having an occult meaning.
For the mathematical meaning of the circle in PA
see a search for "line at infinity."
A different, cubic, model of PA is perhaps more perspicuous.
Heraclitus, Fragment 60 (Diels number):
|
The way up and the way down is one and the same. ὁδὸς ἄνω κάτω μία καὶ ὡυτή hodòs áno káto mía kaì houté |
See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.
Further context: Not Theology.
we are just like a couple of tots…
— Sinatra
Born 1973 in Bergen. Lives and works in Oslo.
Education
2000 – 2004 National Academy of Fine Arts, Oslo
1998 – 2000 Strykejernet Art School, Oslo, NO
1995 – 1998 Philosophy, University of Bergen
University of Bergen—
|
It might therefore seem that the idea of digital and analogical systems as rival fundaments to human experience is a new suggestion and, like digital technology, very modern. In fact, however, the idea is as old as philosophy itself (and may be much older). In his Sophist, Plato sets out the following ‘battle’ over the question of ‘true reality’: What we shall see is something like a battle of gods and giants going on between them over their quarrel about reality [γιγαντομαχία περì της ουσίας] ….One party is trying to drag everything down to earth out of heaven and the unseen, literally grasping rocks and trees in their hands, for they lay hold upon every stock and stone and strenuously affirm that real existence belongs only to that which can be handled and offers resistance to the touch. They define reality as the same thing as body, and as soon as one of the opposite party asserts that anything without a body is real, they are utterly contemptuous and will not listen to another word. (…) Their adversaries are very wary in defending their position somewhere in the heights of the unseen, maintaining with all their force that true reality [την αληθινήν ουσίαν] consists in certain intelligible and bodiless forms. In the clash of argument they shatter and pulverize those bodies which their opponents wield, and what those others allege to be true reality they call, not real being, but a sort of moving process of becoming. On this issue an interminable battle is always going on between the two camps [εν μέσω δε περι ταυτα απλετος αμφοτέρων μάχη τις (…) αει συνέστηκεν]. (…) It seems that only one course is open to the philosopher who values knowledge and truth above all else. He must refuse to accept from the champions of the forms the doctrine that all reality is changeless [and exclusively immaterial], and he must turn a deaf ear to the other party who represent reality as everywhere changing [and as only material]. Like a child begging for 'both', he must declare that reality or the sum of things is both at once [το όν τε και το παν συναμφότερα] (Sophist 246a-249d). The gods and the giants in Plato’s battle present two varieties of the analog position. Each believes that ‘true reality’ is singular, that "real existence belongs only to" one side or other of competing possibilities. For them, difference and complexity are secondary and, as secondary, deficient in respect to truth, reality and being (την αληθινήν ουσίαν, το όν τε και το παν). Difference and complexity are therefore matters of "interminable battle" whose intended end for each is, and must be (given their shared analogical logic), only to eradicate the other. The philosophical child, by contrast, holds to ‘both’ and therefore represents the digital position where the differentiated two yet belong originally together. Here difference, complexity and systematicity are primary and exemplary. It is an unfailing mark of the greatest thinkers of the tradition, like Plato, that they recognize the digital possibility and therefore recognize the principal difference of it from analog possibilities.
— Cameron McEwen, "The Digital Wittgenstein," |
* See that phrase in this journal.
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
From "The Onto-theological Origin of Play:
Heraclitus and Plato," by Yücel Dursun, in
Lingua ac Communitas Vol 17 (October 2007)—
"Heraclitus’s Aion and His Transformations
The saying is as follows:
αἰὼν παῖς ἐστι παίζων, πεττεύων·
παιδὸς ἡ βασιληίη
(Aion is a child playing draughts;
the kingship is the child’s)
(Krell 1972: 64).*
* KRELL, David Farrell.
“Towards an Ontology of Play:
Eugen Fink’s Notion of Spiel,”
Research in Phenomemology ,
2, 1972: 63-93.
This is the translation of the fragment in Greek by Krell.
There are many versions of the translation of the fragment….."
See also Child's Play and Froebel's Magic Box.
Update of May 5— For some background
from the date May 4 seven years ago, see
The Fano Plane Revisualized.
For some background on the word "aion,"
see that word in this journal.
March 28 review in the Times Literary Supplement of
George Steiner's new book The Poetry of Thought—
"If this new book opens with the concession that
language has neither the performative power of music
nor the elegant precision of mathematics,
it is language, for Steiner, that defines the human.
The survey accordingly begins from the ancient Greek
view of man as the 'language-animal.'"
A check of this phrase yields, in a 1969 Steiner essay,
"The Language Animal," a Greek form of the phrase—
In short, the least inadequate definition we can arrive at
of the genus homo , the definition that fully distinguishes
him from all neighbouring life-forms, is this:
man is a zoon phonanta , a language-animal.
— p. 10 in Encounter , August 1969 (essay on pp. 7-23)
After introducing "language-animal" as a translation of "zoon phonanta " in 1969,
Steiner in later writing went on to attribute this phrase to the ancient Greeks.
"The inception of critical thought, of a philosophic anthropology,
is contained in the archaic Greek definition of man as a
'language-animal'…."
— George Steiner, Real Presences , U. of Chicago Press, 1991, p. 89
"… the 'language-animal' we have been since ancient Greece
so designated us…. "
— George Steiner, Grammars of Creation , Yale U. Press, 2002, p. 265
Despite this, there seems to be no evidence for use of this phrase
by the ancient Greeks.
A Google search today for zoon phonanta (ζῷον φωνᾶντα)—
There are also no results from searches for the similar phrases
"ζωον φωναντα," "ζωον φωνᾶντα," and "ζῷον φωναντα."
In memory of director Ulu Grosbard (continued from yesterday)
From http://scripturetext.com/matthew/13-44.htm —
Again the kingdom of heaven is like unto treasure hid in a field
the which when a man hath found he hideth and for joy thereof
goeth and selleth all that he hath and buyeth that field
ΚΑΤΑ ΜΑΤΘΑΙΟΝ 13:44 Greek NT: Byzantine/Majority Text (2000)
παλιν ομοια εστιν η βασιλεια των ουρανων θησαυρω κεκρυμμενω εν τω αγρω
LEXICON —
παλιν adverb
palin pal'-in: (adverbially) anew, i.e. (of place) back, (of time) once more, or (conjunctionally) furthermore or on the other hand — again.
ομοια adjective – nominative singular feminine
homoios hom'-oy-os: similar (in appearance or character) — like, + manner.
εστιν verb – present indicative – third person singular
esti es-tee': he (she or it) is; also (with neuter plural) they are
η definite article – nominative singular feminine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
βασιλεια noun – nominative singular feminine
basileia bas-il-i'-ah: royalty, i.e. (abstractly) rule, or (concretely) a realm — kingdom, + reign.
των definite article – genitive plural masculine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
ουρανων noun – genitive plural masculine
ouranos oo-ran-os': the sky; by extension, heaven (as the abode of God); by implication, happiness, power, eternity; specially, the Gospel (Christianity) — air, heaven(-ly), sky.
θησαυρω noun – dative singular masculine
thesauros thay-sow-ros': a deposit, i.e. wealth — treasure.
κεκρυμμενω verb – perfect passive participle – dative singular masculine
krupto kroop'-to: to conceal (properly, by covering) — hide (self), keep secret, secret(-ly).
εν preposition
en en: in, at, (up-)on, by, etc.
τω definite article – dative singular masculine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
αγρω noun – dative singular masculine
agros ag-ros': a field (as a drive for cattle); genitive case, the country; specially, a farm, i.e. hamlet — country, farm, piece of ground, land.
COLLEGE OF THE DESERT
Minutes — Organization Meeting
11:00 a.m., Saturday, July 1, 1961—
15. Preparation of College Seal:
By unanimous consent preparation of a College
Seal to contain the following features was
authorized: A likeness of the Library building
set in a matrix of date palms, backed by
a mountain skyline and rising sun; before
the Library an open book, the Greek symbol
Alpha on one page and Omega on the other;
the Latin Lux et Veritas, College of the
Desert, and 1958 to be imprinted within or
around the periphery of the seal.
From the website http://geofhagopian.net/ of
Geoff Hagopian, Professor of Mathematics,
College of the Desert—
Note that this version of the seal contains
an Aleph and Omega instead of Alpha and Omega.
From another Hagopian website, another seal.
From Rebecca Goldstein's Talks and Appearances page—
• "36 (Bad) Arguments for the Existence of God,"
Annual Meeting of the Freedom from Religion Foundation,
Marriot, Hartford, CT, Oct 7 [2011], 7 PM
From Wallace Stevens—
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
For those who prefer greater depth on Yom Kippur, yesterday's cinematic link suggests…
"Yo sé de un laberinto griego que es una línea única, recta."
—Borges, "La Muerte y la Brújula " ("Death and the Compass")
See also Alpha and Omega (Sept. 18, 2011) and some context from 1931.
For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)
Plato's Diamond
The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—
Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate…. … Both in the Republic and in the Sophist there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno that the square on the diagonal is double the original square in area." – William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985 Plato's paradigm in the Meno—
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change—
Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |
Related material: Plato's Diamond by Oslo artist Josefine Lyche.
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"Remember me to her."
— Closing words of the Algis Budrys novel Rogue Moon .
Background— Some posts in this journal related to Abel or to random thoughts from his birthday.
The title was suggested by this evening's 4-digit NY lottery number.
"… the rhetoric might be a bit over the top."
According to Amazon.com, 2198 (i.e., 2/1/98) was the publication
date of Geometry of Vector Sheaves , Volume I, by Anastasios Mallios.
Related material—
The question of S.S. Chern quoted here June 10: —
"What is Geometry?"— and the remark by Stevens that
accompanied the quotation—
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— Wallace Stevens,
“An Ordinary Evening in New Haven” VI
The work of Mallios in pure mathematics cited above seems
quite respectable (unlike his later remarks on physics).
His Vector Sheaves appears to be trying to explore new territory;
hence the relevance of Stevens's "Alpha." See also the phrase
"A-Invariance" in an undated preprint by Mallios*.
For the evening 3-digit number, 533, see a Stevens poem—
This meditation by Stevens is related to the female form of Mallios's Christian name.
As for the afternoon numbers, see "62" in The Beauty Test (May 23, 2007), Geometry and Death, and "9181" as the date 9/1/81.
* Later published in International Journal of Theoretical Physics , Vol. 47, No. 7, cover date 2008-07-01
Some background for yesterday’s posts:
Midrash for Gnostics and related notes,
as well as yesterday’s New York Lottery.
…. “We seek
The poem of pure reality, untouched
By trope or deviation, straight to the word,
Straight to the transfixing object, to the object
At the exactest point at which it is itself,
Transfixing by being purely what it is….”
— Wallace Stevens (1879-1955),
“An Ordinary Evening in New Haven” IX
“Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals.”
— Wallace Stevens,
“An Ordinary Evening in New Haven” VI
“A hierophant is a person who brings religious congregants into the presence of that which is deemed holy . The word comes from Ancient Greece, where it was constructed from the combination of ta hiera , ‘the holy,’ and phainein , ‘to show.’ In Attica it was the title of the chief priest at the Eleusinian Mysteries. A hierophant is an interpreter of sacred mysteries and arcane principles.”
Weyl as Alpha, Chern as Omega—
Postscript for Ellen Page, star of “Smart People”
and of “X-Men: The Last Stand“— a different page 679.
Your assignment, should you choose to accept it—
Interpret today’s NY lottery numbers— Midday 815, Evening 888.
My own bias is toward 815 as 8/15 and 888 as a trinity,
but there may be less obvious and more interesting approaches.
"These passages suggest that the Form is a character or set of characters
common to a number of things, i.e. the feature in reality which corresponds
to a general word. But Plato also uses language which suggests not only
that the forms exist separately (χωριστά ) from all the particulars, but also
that each form is a peculiarly accurate or good particular of its own kind,
i.e. the standard particular of the kind in question or the model (παράδειγμα )
[i.e. paradigm ] to which other particulars approximate….
… Both in the Republic and in the Sophist there is a strong suggestion
that correct thinking is following out the connexions between Forms.
The model is mathematical thinking, e.g. the proof given in the Meno
that the square on the diagonal is double the original square in area."
— William and Martha Kneale, The Development of Logic,
Oxford University Press paperback, 1985
Plato's paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change* —
Monochrome figures to
colored figures
Areas to
transformations
Continuous transformations to
non-continuous transformations
Euclidean geometry to
finite geometry
Euclidean quantities to
finite fields
Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.
* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα above, along with the commentators' specific reference to
the Meno figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.
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