Kernel of
Eternity:
Notes from the Journal of
Steven H. Cullinane,
June 19-21, 2007
Introduction
(Aug. 13, 2007)
Adam Gopnik in
The New Yorker of
August 20, 2007--
On
Philip K. Dick:
"... the kind of guy who can’t drink one cup of coffee
without drinking six,
and then stays up all night to tell you what Schopenhauer really said
and how it affects your understanding of Hitchcock and what that had to
do with Christopher Marlowe."
Modernity:
A Film by
Alfred Hitchcock--
"... the most thoroughgoing modernist design element in
Hitchcock's films arises out of geometry, as François Regnault
has argued, identifying 'a global movement for each one, or a "principal
geometric or dynamic form," which can appear in the pure
state in the credits....'" --Peter J. Hutchings (my italics)
Thursday, June 21, 2007 4:30 PM
Found in translation:
"Ich
aber, hier auf dem objektiven Wege, bin jetzt bemüht, das Positive
der
Sache nachzuweisen, daß nämlich das Ding an sich von der
Zeit und Dem,
was nur durch sie möglich ist, dem Entstehen und Vergehen,
unberührt
bleibt, und daß die Erscheinungen in der Zeit sogar jenes rastlos
flüchtige, dem Nichts zunächst stehende Dasein nicht haben
könnten,
wenn nicht in ihnen ein Kern aus der Ewigkeit*
wäre. Die Ewigkeit ist freilich ein Begriff, dem keine Anschauung
zum
Grunde liegt: er ist auch deshalb bloß negativen Inhalts, besagt
nämlich ein zeitloses Dasein. Die Zeit ist demnach ein
bloßes Bild der
Ewigkeit, ho chronos eikôn tou aiônos,**
wie es Plotinus***
hat: und ebenso ist unser zeitliches Dasein das bloße Bild unsers
Wesens an sich. Dieses muß in der Ewigkeit liegen, eben weil die
Zeit
nur die Form unsers Erkennens ist: vermöge dieser allein aber
erkennen
wir unser und aller Dinge Wesen als vergänglich, endlich und der
Vernichtung anheimgefallen."
* "a kernel of eternity"
** "Time is the image of eternity."
*** "wie es Plotinus hat"--
Actually, not Plotinus, but Plato,
according to Diogenes Laertius.
Related material:
Time
Fold,
J. N. Darby,
"On the Greek Words for
Eternity and Eternal
(aion and aionios),"
Carl Gustav Jung, Aion,
which contains the following
four-diamond figure,
and Jung
and the Imago Dei.
Thursday, June 21, 2007 12:07 PM
Structural Logic continued:
"His graceful accounts of the Bach Suites for Unaccompanied Cello
illuminated the works’ structural logic as well as their inner
spirituality."
--Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007
"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
"...
the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it. (It
induces A8 on the octad, the kernel
of this action being the translation group of the affine space.)"
-- Peter J. Cameron, "The
Geometry of the Mathieu Groups" (pdf)
"... donc Dieu existe,
réponse!"
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two, as
quoted by Jung in
Memories,
Dreams, Reflections)
"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
-- Physics Today
"Borja
dropped the mutilated book on the floor with the others. He was looking
at the nine engravings and at the circle, checking strange
correspondences between them.
'To meet someone' was his
enigmatic answer. 'To search for the stone that the Great Architect
rejected, the philosopher's stone, the basis of the philosophical work.
The stone of power. The devil likes metamorphoses, Corso.'"
-- The Club Dumas, basis for the Roman Polanski
film "The Ninth Gate" (See 12/24/05.)
"Pauli linked this symbolism
with the concept of automorphism."
-- The Innermost Kernel
(previous entry)
And from
"Symmetry in Mathematics
and Mathematics of Symmetry"
(pdf), by Peter J. Cameron,
a paper presented at the
International
Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have
The Epigraph--
(Here "whatever" should
of course be "whenever.")
Also from the
Cameron paper:
Local or global?
Among other (mostly more vague) definitions of symmetry, the
dictionary will typically list two, something like this:
• exact correspondence of parts;
• remaining unchanged by transformation.
Mathematicians
typically consider the second, global, notion, but what about the
first, local, notion, and what is the relationship between them?
A
structure M is homogeneous if every isomorphism between
finite substructures of M can be extended to an automorphism of
M; in other words, "any local symmetry is global."
|
Some Log24 entries
related to the above politically
(women in mathematics)--
Global and Local:
One Small Step
and mathematically--
Structural Logic continued:
Structure and Logic (4/30/07):
This entry cites
Alice
Devillers of Brussels--
"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
as many symmetries as possible."
"There is such a thing
as a tesseract."
-- Madeleine L'Engle
Wednesday, June 20, 2007 1:06 AM
ART WARS continued:
Kernel
Mathematical
Reviews citation:
MR2163497 (2006g:81002) 81-03
(81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and
quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag,
Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9
A quote from MR at
Amazon.com:
"This revised translation of a Swedish Ph. D. thesis in philosophy
offers far more than a discussion of Wolfgang Pauli's encounters with
the psychoanalyst Carl Gustav Jung.... Here the book explains very well
how Pauli attempted to extend his understanding beyond superficial
esotericism and spiritism.... To understand Pauli one needs books like
this one, which... seems to open a path to a fuller understanding of
Pauli, who was seeking to solve a quest even deeper than quantum
physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
An excerpt:
I
do not yet know what Gieser means by "the innermost kernel." The
following is my version of a "kernel" of sorts-- a diagram well-known
to students of anthropologist Claude Levi-Strauss and art theorist Rosalind
Krauss:
The four group is also known as the Vierergruppe
or Klein group. It appears, notably, as the translation subgroup
of A, the group of 24 automorphisms of the affine plane over
the 2-element field, and therefore as the kernel of the
homomorphism taking A to the group of 6 automorphisms of the
projective line over the 2-element field. (See Finite Geometry of
the Square and Cube.)
Tuesday, June 19, 2007 3:17 PM
Meta Physics continued:
Faustus
is gone:
regard his hellish fall
I have just read, in the New
York Times Book Review that arrived in yesterday's mail, a review
of Segre's Faust in Copenhagen. The review, on
news stands next Sunday, was titled by the Times "Meta
Physicists."
On Faust-- today's noon entry and yesterday's "Nightmare Lessons."
On "Meta Physicists"-- an entry of June 6, on Cullinane College, has a section
titled "Meta Physics."
On Copenhagen-- an entry of Bloomsday Eve, 2004 on a native of that city.
Another Dane:
"Words, words, words."
-- Hamlet
Another metaphysics:
"317 is a prime,
not because we think so,
or because our minds
are shaped in one way
rather than another,
but because it is so,
because mathematical
reality is built that way."
-- G. H. Hardy,
A Mathematician's Apology
Epilogue
(Aug. 13, 2007)
Adam Gopnik is also the author
of
The King in the Window, a tale
of the Christian feast of Epiphany
and a sinister quantum computer.
For more on Epiphany, see
the Log24 entries of August 1.
For more on quantum computing,
see What is Quantum Computation?.
See also the previous entry
("The Geometry of Qubits,"
Aug. 12, 2007).
Sunday, August 12, 2007 9:00 AM
In the context of
quantum
information theory, the following structure seems to be of
interest--
"... the full two-by-two matrix ring with entries in
GF(2),
M2(
GF(2))--
the unique simple non-commutative ring of order 16 featuring six units
(invertible elements) and ten zero-divisors."
-- "Geometry of Two-Qubits," by Metod Saniga (
pdf,
17 pp.), Jan. 25, 2007
This ring is another way of looking at the 16
elements of the affine space
A4(
GF(2))
over the 2-element field. (Arrange the four coordinates of each
element-- 1's and 0's-- into a square instead of a straight line, and
regard the resulting squares as matrices.) (For more on
A4(
GF(2)),
see
Finite Relativity and related notes at
Finite
Geometry of the Square and Cube.) Using the above ring,
Saniga constructs a system of 35 objects (not unlike the 35 lines of
the
finite geometry PG(3,2)) that he calls a "projective line"
over the ring. This system of 35 objects has a subconfiguration
isomorphic to
the (2,2) generalized quadrangle W2
(which occurs naturally as a subconfiguration of
PG(3,2)-- see
Inscapes.)
Saniga concludes:
"We have demonstrated that the basic properties of
a system of two interacting spin-1/2 particles are uniquely embodied in
the (sub)geometry of a particular projective line, found to be
equivalent to the generalized quadrangle of order two. As such systems
are the simplest ones exhibiting phenomena like quantum entanglement
and quantum non-locality and play, therefore, a crucial role in
numerous applications like quantum cryptography, quantum coding,
quantum cloning/teleportation and/or quantum computing to mention the
most salient ones, our discovery thus
-
not only offers a principally new
geometrically-underlined insight into their intrinsic nature,
-
but also gives their applications a wholly new
perspective
-
and opens up rather unexpected vistas for an
algebraic geometrical modelling of their higher-dimensional
counterparts."
is not without relevance to
the physics of quantum theory.
