From a post on St. Andrew's Day, 2017 —
See also "ENumbers" and "EGirls."
See a Quanta Magazine article published today and
https://www.chapman.edu/ourfaculty/mattleifer.
From the article —
Leifer, for his part, is holding out for something new. “I think the correct interpretation of quantum mechanics is none of the above,” he said. He likens the current situation with quantum mechanics to the time before Einstein came up with his special theory of relativity. Experimentalists had found no sign of the “luminiferous ether” — the medium through which light waves were thought to propagate in a Newtonian universe. Einstein argued that there is no ether. Instead he showed that space and time are malleable. “PreEinstein I couldn’t have told you that it was the structure of space and time that was going to change,” Leifer said. Quantum mechanics is in a similar situation now, he thinks. “It’s likely that we are making some implicit assumption about the way the world has to be that just isn’t true,” he said. “Once we change that, once we modify that assumption, everything would suddenly fall into place. That’s kind of the hope. Anybody who is skeptical of all interpretations of quantum mechanics must be thinking something like this. Can I tell you what’s a plausible candidate for such an assumption? Well, if I could, I would just be working on that theory.” 
See as well this journal on the Feast of the Assumption, 2018.
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss, 1976
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
From a Log24 post of June 2627, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
"An awful lot of important dualities in four and fewer dimensions
follow from this sixdimensional theory and its properties."
— Edward Witten, interviewed by Natalie Wolchover,
in Quanta Magazine on November 28, 2017
See also Six Dimensions in this journal.
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.
American Songwriters (the Petrusich versions)
The New Yorker Culture Desk online yesterday —
The New Yorker Culture Desk onine July 16 —
Related aesthetic meditation from this journal on July 16 —
(From the post "Schoolgirl Space for Quantum Mystics")
See also Clifford in this journal, in particular
The Matrix for Quantum Mystics
(Log24, St. Andrew's Day, 2017).
Some context for what Heidegger called
das SpiegelSpiel des Gevierts
From Helen Lane's translation of El Mono Gramático ,
a book by Nobel winner Octavio Paz first published
in Barcelona by Seix Barral in 1974 —
Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oftrepeated and everdifferent metaphor of identity.
— Paz, Octavio. The Monkey Grammarian 
A related 1960 meditation from Claude LéviStrauss taken from a
Log24 post of St. Andrew's Day 2017, "The Matrix for Quantum Mystics":
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
The title of the previous post, "For Quantum Mystics,"
suggests a search in this journal for Quantum + Mystic.
That search in turn suggests, in particular, a review of
a post of October 16, 2007 — a discussion of the
P.T. Barnumlike phrase "deep beauty" used to describe
a topic under discussion at Princeton by physicists.
Princeton, by the way, serves to illustrate the "gutter"
mentioned by Sir Laurence Olivier in a memorable
classroom scene from 1962.
As a corrective to the previous parodies here, the following material on the mathematician HarishChandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.
MacTutor History of Mathematics:
“HarishChandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBMvon Neumann Professor in 1968.”
R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):
“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semisimple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”
“It is difficult to communicate the grandeur of HarishChandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”
For deeper views of his work, see
From The New York Times on June 20, 2018 —
" In a widely read article published early this year on arXiv.org,
a site for scientific papers, Gary Marcus, a professor at
New York University, posed the question:
'Is deep learning approaching a wall?'
He wrote, 'As is so often the case, the patterns extracted
by deep learning are more superficial than they initially appear.' "
See as well an image from posts tagged Quantum Suffering . . .
The time above, 10:06:48 PM July 16, is when I saw …
"What you mean 'we,' Milbank?"
A New, Improved Version of Quantum Suffering !
Background for group actions on the eightfold cube —
See also other posts now tagged Quantum Suffering
as well as — related to the image above of the Great Wall —
The Quantum Tesseract Theorem —
Raiders —
A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling
Time Magazine December 25, 2017 – January 1, 2018
A figure related to the general connecting theorem of Koen Thas —
See also posts tagged Dirac and Geometry in this journal.
Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."
TIME magazine, issue of December 25th, 2017 —
" In 2003, Hand worked with Disney to produce a madeforTV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "
Not the dream? It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.
From a trailer for the latest version —
Detail:
From the 1962 book —
"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."
Song adapted from a 1960 musical —
"In short, there's simply not
A more congenial spot
For happyeveraftering
Than here in Camazotz!"
The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013. The talk's subtitle was
"Oswald Veblen and the Sixhundredacre Woods."
Meanwhile…
Thursday, March 21, 2013

Related material for those who prefer narrative
to mathematics:
Log24 on June 6, 2006:
The Omen :

Related material for those who prefer mathematics
to narrative:
What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry. As the British
geometer H. F. Baker pointed out, 6 is a rather important number
in that discipline. For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.
See also last night's Veblen and Young in Light of Galois.
"There is such a thing as a tesseract." — Madeleine L'Engle
The following picture provides a new visual approach to
the order8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she cofounded—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
The Principle of Sufficient Reason
from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940
EXCERPT 1—
My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.^{2}
In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
“Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”
If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multidimensionally^{3} whereas men can only
think in linear syllogistic series, and the Theory of Groups is
^{2} As far as I am aware, only Scholastic Philosophy has fully recognized and ex
ploited this principle as one of basic importance for philosophic thought
^{3} That is, uses multidimensional symbols beyond our grasp.
______________________________________________________________________
the appropriate instrument of thought to remedy our deficiency
in this respect.
The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,^{4} “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.^{5}
Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.
Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D; B, C, D, A; C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a
^{4} My translation.
^{5} It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x^{2}6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________
null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________
EXCERPT 2—
Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasimathematical form as follows:
PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).
HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly selfevident in virtue of the above Principle.
The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.
In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.
It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu
sion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.
___________________________________________________________________________
A more recent lecture on the same subject —
by JeanPierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)
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