Log24

Thursday, October 22, 2015

Objective Quality

Filed under: General,Geometry — Tags: — m759 @ 2:26 AM

Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:

A more interesting account of these 35 structures:

"It is commonly known that there is a bijection between
the 35 unordered triples of a 7-set [i.e., the 35 partitions
of an 8-set into two 4-sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.

For some context, see Eightfold Geometry by Steven H. Cullinane.

Saturday, September 17, 2011

Objectivity

Filed under: General — m759 @ 12:00 PM

The previous two posts, Baggage and The Uploading, suggest
a review of Wroclaw's native son Ernst Cassirer.

Wednesday, August 10, 2011

Objectivity

Filed under: General,Geometry — m759 @ 12:25 PM

From math16.com

Quotations on Realism
and the Problem of Universals:

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268

"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.

"People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only 'truths' strictly worthy of the name. Such truths I will call 'diamonds'; they are highly desirable but hard to find….The happy metaphor is Morris Kline's in Mathematics in Western Culture (Oxford, 1953), p. 430."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 114 and 117

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call 'models of reality'). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 256 and 259

Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is non-Euclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000

The correct statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)— 

From The Search for Invariants (June 20, 2011):

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

http://www.log24.com/log/pix11B/110810-MajaLovrenovBio.jpg

Related material from Sunday's New York Times  travel section—

"Exhibit A is certainly Ljubljana…."

Wednesday, March 6, 2019

The Relativity Problem and Burkard Polster

Filed under: General,Geometry — Tags: — m759 @ 11:28 AM
 

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

For some context, see Relativity Problem  in this journal.

In the case of PG(3,2), there is a choice of geometric models 
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.

The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.

For comparison of the two models, see a figure posted here on
May 21, 2014 —

Labeling the Tetrahedral Model  (Click to enlarge) —

"Citation needed" —

The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)
," a "reliable source."

Monday, December 3, 2018

The Relativity Problem at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:21 PM

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

See also Relativity Problem and Diamonds and Whirls.

Wednesday, October 17, 2018

Aesthetics

Filed under: General,Geometry — Tags: — m759 @ 11:22 AM
 

From "The Phenomenology of Mathematical Beauty,"
by Gian-Carlo Rota —

The Lightbulb Mistake

. . . . Despite the fact that most proofs are long, and despite our need for extensive background, we think back to instances of appreciating mathematical beauty as if they had been perceived in a moment of bliss, in a sudden flash like a lightbulb suddenly being lit. The effort put into understanding the proof, the background material, the difficulties encountered in unraveling an intricate sequence of inferences fade and magically disappear the moment we become aware of the beauty of a theorem. The painful process of learning fades from memory, and only the flash of insight remains.

We would like  mathematical beauty to consist of this flash; mathematical beauty should  be appreciated with the instantaneousness of a lightbulb being lit. However, it would be an error to pretend that the appreciation of mathematical beauty is what we vaingloriously feel it should be, namely, an instantaneous flash. Yet this very denial of the truth occurs much too frequently.

The lightbulb mistake is often taken as a paradigm in teaching mathematics. Forgetful of our learning pains, we demand that our students display a flash of understanding with every argument we present. Worse yet, we mislead our students by trying to convince them that such flashes of understanding are the core of mathematical appreciation.

Attempts have been made to string together beautiful mathematical results and to present them in books bearing such attractive titles as The One Hundred Most Beautiful Theorems of Mathematics . Such anthologies are seldom found on a mathematician’s bookshelf. The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory. But when mathematical theorems from disparate areas are strung together and presented as “pearls,” they are likely to be appreciated only by those who are already familiar with them.

The Concept of Mathematical Beauty

The lightbulb mistake is our clue to understanding the hidden sense of mathematical beauty. The stark contrast between the effort required for the appreciation of mathematical beauty and the imaginary view mathematicians cherish of a flashlike perception of beauty is the Leitfaden  that leads us to discover what mathematical beauty is.

Mathematicians are concerned with the truth. In mathematics, however, there is an ambiguity in the use of the word “truth.” This ambiguity can be observed whenever mathematicians claim that beauty is the raison d’être of mathematics, or that mathematical beauty is what gives mathematics a unique standing among the sciences. These claims are as old as mathematics and lead us to suspect that mathematical truth and mathematical beauty may be related.

Mathematical beauty and mathematical truth share one important property. Neither of them admits degrees. Mathematicians are annoyed by the graded truth they observe in other sciences.

Mathematicians ask “What is this good for?” when they are puzzled by some mathematical assertion, not because they are unable to follow the proof or the applications. Quite the contrary. Mathematicians have been able to verify its truth in the logical sense of the term, but something is still missing. The mathematician who is baffled and asks “What is this good for?” is missing the sense  of the statement that has been verified to be true. Verification alone does not give us a clue as to the role of a statement within the theory; it does not explain the relevance  of the statement. In short, the logical truth of a statement does not enlighten us as to the sense of the statement. Enlightenment , not truth, is what the mathematician seeks when asking, “What is this good for?” Enlightenment is a feature of mathematics about which very little has been written.

The property of being enlightening is objectively attributed to certain mathematical statements and denied to others. Whether a mathematical statement is enlightening or not may be the subject of discussion among mathematicians. Every teacher of mathematics knows that students will not learn by merely grasping the formal truth of a statement. Students must be given some enlightenment as to the sense  of the statement or they will quit. Enlightenment is a quality of mathematical statements that one sometimes gets and sometimes misses, like truth. A mathematical theorem may be enlightening or not, just as it may be true or false.

If the statements of mathematics were formally true but in no way enlightening, mathematics would be a curious game played by weird people. Enlightenment is what keeps the mathematical enterprise alive and what gives mathematics a high standing among scientific disciplines.

Mathematics seldom explicitly acknowledges the phenomenon of enlightenment for at least two reasons. First, unlike truth, enlightenment is not easily formalized. Second, enlightenment admits degrees: some statements are more enlightening than others. Mathematicians dislike concepts admitting degrees and will go to any length to deny the logical role of any such concept. Mathematical beauty is the expression mathematicians have invented in order to admit obliquely the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of this phenomenon. They say that a theorem is beautiful when they mean to say that the theorem is enlightening. We acknowledge a theorem’s beauty when we see how the theorem “fits” in its place, how it sheds light around itself, like Lichtung — a clearing in the woods. We say that a proof is beautiful when it gives away the secret of the theorem, when it leads us to perceive the inevitability of the statement being proved. The term “mathematical beauty,” together with the lightbulb mistake, is a trick mathematicians have devised to avoid facing up to the messy phenomenon of enlightenment. The comfortable one-shot idea of mathematical beauty saves us from having to deal with a concept that comes in degrees. Talk of mathematical beauty is a cop-out to avoid confronting enlightenment, a cop-out intended to keep our description of mathematics as close as possible to the description of a mechanism. This cop-out is one step in a cherished activity of mathematicians, that of building a perfect world immune to the messiness of the ordinary world, a world where what we think should be true turns out to be true, a world that is free from the disappointments, ambiguities, and failures of that other world in which we live.

How many mathematicians does  it take to screw in a lightbulb?

Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-160917.jpg

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-161018.jpg

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of.

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space  coordinates. He describes it as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

http://www.log24.com/log/pix18/180827-MIT-Rubik-Robot.jpg

Monday, August 13, 2018

Trojan Horsitude

Filed under: General,Geometry — Tags: — m759 @ 3:33 AM
 

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."

— Simon Blackburn, Think,
    Oxford University Press, 1999, page 268


". . . a perfect triptych of horsitude"

James Parker on the 2007 film "Michael Clayton"

Related material —

Horsitude in the 4×2 grid, and

http://www.log24.com/log/pix18/180813-Structure_and_Sense-post-160606.gif

http://www.log24.com/log/pix18/180813-Knight_Moves-080116-page-top.gif

Wednesday, April 25, 2018

An Idea

Filed under: General,Geometry — m759 @ 11:45 AM

"There was an idea . . ." — Nick Fury in 2012

". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."

New York Times  dance critic Alastair Macaulay,
    quoted here in a post of August 20, 2011.

Illustration from that post —

A 2x4 array of squares

See also Macaulay in
last night's 10 PM post.

Tuesday, January 2, 2018

Debs and Redhead

Filed under: General — m759 @ 3:15 PM

Or:  Schoolgirl Problems

The above images were suggested in part by the birthdays
on Sept. 21, 2011, of Bill Murray and Stephen King.

More seriously, also in this journal on that date, from a post
titled Symmetric Generation —

Thursday, March 23, 2017

More Harvard Ignorance

Filed under: General — Tags: — m759 @ 6:42 PM

"… the leftist war on truth, the never-ending campaign
to recast objective fact as subjective and open to question."

— Kyle Smith in The New Criterion  on March 18

"A sort of flint stone" —

See also the above six-part image in the previous post.

Yabba Dabba Doo.

Monday, March 20, 2017

December 1987 at Yale

Filed under: General — m759 @ 9:29 PM

"I was at the time a Yale English major (we read, appreciated,
and discussed the meaning of literature) sunk in the toxic quagmire
of the one and only course I ever took in the literature department
(where authorial intent was ignored and every 'text' was considered
solely on how comfortably it nestled within the shackles of Marxism)."

"For decades de Man had been an avatar not just of leftist politics
but also of the leftist war on truth, the never-ending campaign
to recast objective fact as subjective and open to question."

Kyle Smith in The New Criterion  on March 18

See as well other posts mentioning Kyle Smith in this  journal.

Wednesday, March 8, 2017

Inscapes

Filed under: General,Geometry — Tags: — m759 @ 6:42 PM

"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

Illustration

Commentary

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985.

Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964.

Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982.

Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Sunday, February 26, 2017

Transformers Meet Transformations

Filed under: General — m759 @ 10:30 AM

"Transformations , Anne Sexton’s 1971 collection of poems, is a portal."

— "A Poisonous Antidote," by Nick Ripatrazone, at themillions.com
      at noon on October 22, 2015

"You see, opening dimensional portals is a tricky business."

— The librarian in "Buffy the Vampire Slayer," season 1, episode 2

See also Transformers in this journal.

Synchronology

"Objective Quality" in this journal on the date of the above review,
October 22, 2015, at 2:26 AM ET.

Friday, February 17, 2017

Code

Filed under: General — m759 @ 7:00 PM

Steve Martin on his character Ray Porter 
in the novella Shopgirl  —

http://www.log24.com/log/pix10B/101122-MartinShopgirl-loq.jpg

"He said, 'I wrote a piece of code
 that they just can’t seem to do without.'
 He was a symbolic logician. That was his career…."

"In short, he seeks to objectify crucial areas . . . ."

Or crucify objective areas.

Sunday, February 12, 2017

Colorful Tales

Filed under: General — m759 @ 1:23 PM

“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”

— Hermann Weyl, Philosophy of  Mathematics and
    Natural Science 
, Princeton, 1949, p. 237

Harvard University Press on the late Angus Fletcher, author of
The Topological Imagination  and Colors of the Mind

From the Harvard webpage for Colors of the Mind

Angus Fletcher is one of our finest theorists of the arts,
the heir to I. A. Richards, Erich Auerbach, Northrop Frye.
This… book…  aims to open another field of study:
how thought— the act, the experience of thinking—
is represented in literature.

. . . .

Fletcher’s resources are large, and his step is sure.
The reader samples his piercing vision of Milton’s

Satan, the original Thinker,
leaving the pain of thinking
as his legacy for mankind.

A 1992 review by Vinay Dharwadker of Colors of the Mind —

See also the above word "dianoia" in The Echo in Plato's Cave.
Some context 

This post was suggested by a memorial piece today in
the Los Angeles Review of Books

A Florilegium for Angus Fletcher

By Kenneth Gross, Lindsay Waters, V. N. Alexander,
Paul Auster, Harold Bloom, Stanley Fish, K. J. Knoespel,
Mitchell Meltzer, Victoria Nelson, Joan Richardson,
Dorian Sagan, Susan Stewart, Eric Wilson, Michael Wood

Fletcher reportedly died on November 28, 2016.

"I learned from Fletcher how to apprehend
the daemonic element in poetic imagination."

— Harold Bloom in today's Los Angeles florilegium

For more on Bloom and the daemonic, see a Log24 post,
"Interpenetration," from the date of Fletcher's death.

Some backstory:  Dharwadker in this journal.

Thursday, December 1, 2016

Correlation/Correlative

Filed under: General — m759 @ 6:00 AM

http://m759.net/wordpress/?s="Correlation"

http://m759.net/wordpress/?tag=correlative

Related literary reference —

"The only way of expressing emotion in the form of art
is by finding an 'objective correlative'; in other words,
a set of objects, a situation, a chain of events which
shall be the formula of that particular  emotion; such that
when the external facts, which must terminate in sensory
experience, are given, the emotion is immediately evoked.
If you examine any of Shakespeare’s more successful
tragedies, you will find this exact equivalence…."

— T. S. Eliot, "Hamlet and His Problems" (1919)

Tuesday, November 22, 2016

Jargon

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM

See "sacerdotal jargon" in this journal.

For those who prefer scientific  jargon —

"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

For some science that is not just jargon, see

and, also from posts tagged Dirac and Geometry

Anticommuting Dirac matrices as spreads of projective lines

The above line complex also illustrates an outer automorphism
of the symmetric group S6. See last Thursday's post "Rotman and
the Outer Automorphism
."

Tuesday, October 18, 2016

Parametrization

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

Monday, May 2, 2016

Quality

Filed under: General,Geometry — Tags: — m759 @ 3:48 PM

The previous post, on subjective  and objective  quality,
suggests a review of Pirsig

     "And finally: Phaedrus, following a path
that to his knowledge had never been taken before
in the history of Western thought,
went straight between the horns of
the subjectivity-objectivity dilemma and said
Quality is neither a part of mind, nor is it a part of matter.
It is a third  entity which is independent of the two.
     He was heard along the corridors
and up and down the stairs of Montana Hall
singing softly to himself, almost under his breath,
'Holy, holy, holy…blessed Trinity.' "

See also Guitart in this journal, noting esp. Zen and the Art.

Subjective Quality

Filed under: General,Geometry — m759 @ 6:01 AM

The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.

Siobhan Roberts recently wrote a book about the first of these
authors, Conway.  I just discovered that last fall she also had an
article about the second author, Sloane, published:

"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.

Meanwhile, in this  journal

Log24 on that same date, Oct. 22, 2015 —

Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective  quality, as opposed to the objective  quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here  last October.

Thursday, October 22, 2015

A Marriage of Heaven and Hell

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

Later … (Click to enlarge.)

See as well last night's post Objective Quality.

Thursday, April 23, 2015

Colorful Tale

Filed under: General,Geometry — m759 @ 12:00 PM

(A sequel to yesterday's ART WARS and this
morning's De Colores )

“Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind– as Eddington puts it– the colorful tale
of the subjective storyteller mind.” — Hermann Weyl
(Philosophy of  Mathematics and Natural Science ,
Princeton, 1949, p. 237)

See also Deathly Hallows.

Friday, March 27, 2015

Pursuit of Gestalt*

Filed under: General — m759 @ 12:00 PM

The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.

Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.

Gestalt-related material from this journal that week —

* Title suggested by that of a book by Quine.

Monday, February 2, 2015

Spielraum as Ω

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM
 

From "Origins of the Logical Theory of Probability: von Kries, Wittgenstein, Waismann," by Michael Heidelberger —

"Von Kries calls a range of objective possibilities of a hypothesis or event (under given laws) its Spielraum   (literally: play space), which can mean ‘room to move’, ‘leeway’, ‘latitude of choice’, ‘degree of freedom’ or ‘free play’ and ‘clearance’ – or even ‘scope’. John Maynard Keynes translated it as ‘field’, but the term ‘range’ has generally been adopted in English. Von Kries now holds that if numerical probability were to make any sense at all it must be through this concept of the Spielraum  . Von Kries’s theory is therefore called a ‘Spielraum  theory’ or ‘range theory of probability’."

— International Studies in the Philosophy of Science , Volume 15, Issue 2, 2001, pp. 177-188

See also the tag Points Omega
(Scroll down to January 11-12, 2015.)

Related material:

"Now, for example, in how far are
the six sides of a symmetric die
'equally possible' upon throwing?"

— From "The Natural-Range Conception
     of Probability," by Dr. Jacob Rosenthal,
     page 73 in Time, Chance, and
     Reduction: Philosophical Aspects of
     Statistical 
Mechanics , ed. by 
     Gerhard Ernst and Andreas Hüttemann, 
     Cambridge U. Press, 2010, pp. 71-90

Tuesday, December 2, 2014

Colorful Tale

Filed under: General — m759 @ 9:45 AM

Continued.

"Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind— as Eddington puts it— the colorful tale
of the subjective storyteller mind."

— Hermann Weyl in Philosophy of Mathematics
     and Natural Science
 , Princeton, 1949, p. 237

Tom Wolfe on art theorists in The Painted Word  (1975) :

"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene."

The Weyl quote is a continuing theme in this journal.
The Wolfe quote appeared here on Nov. 18, 2014,
the reported date of death of Yale graduate student 
Natasha Chichilnisky-Heal.

Directions to her burial (see yesterday evening) include
a mention of "Paul Robson Street" (actually Paul
Robeson Place) near "the historic Princeton Cemetery."

This, together with the remarks by Tom Wolfe posted
here on the reported day of her death, suggests a search
for "red green black" —

The late Chichilnisky-Heal was a student of political economy.

The search colors may be interpreted, if one likes, as referring
to politics (red), economics (green), and Robeson (black).

See also Robeson in this journal.

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Wednesday, January 30, 2013

Abstract Possibility

Filed under: General — m759 @ 1:01 PM

Today's NY Times  "Stone Links" to philosophy include
a link to a review of a collection of Hilary Putnam's papers.

Related material, from Putnam's "What is Mathematical
Truth?
" (Historia Mathematica  2 (1975): 529-543)—

"In this paper I argue that mathematics should be interpreted realistically – that is, that mathematics makes assertions that are objectively true or false, independently of the human mind, and that something answers to such mathematical notions as ‘set’ and ‘function’. This is not to say that reality is somehow bifurcated – that there is one reality of material things, and then, over and above it, a second reality of ‘mathematical things’. A set of objects, for example, depends for its existence on those objects: if they are destroyed, then there is no longer such a set. (Of course, we may say that the set exists ‘tenselessly’, but we may also say the objects exist ‘tenselessly’: this is just to say that in pure mathematics we can sometimes ignore the important difference between ‘exists now’ and ‘did exist, exists now, or will exist’.) Not only are the ‘objects’ of pure mathematics conditional upon material objects; they are, in a sense, merely abstract possibilities. Studying how mathematical objects behave might better be described as studying what structures are abstractly possible and what structures are not abstractly possible."

See also Wittgenstein's Diamond and Plato's Diamond.

Thursday, December 27, 2012

Object Lesson

Filed under: General,Geometry — Tags: — m759 @ 3:17 AM

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry—

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

IMAGE- Ernst Cassirer on 'representation' or 'Vorstellung' in 'Substance and Function' as 'the riddle of knowledge'

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

Sunday, December 23, 2012

In a Nutshell…

Filed under: General — m759 @ 1:00 AM

The Kernel of the Concept of the Object

according to the New York Lottery yesterday—

From 4/27

From 11/24

IMAGE- Agent Smith from 'The Matrix,' 1999

A page numbered 176

A page numbered 187

Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — m759 @ 10:48 PM

(Continued from 1986)

S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: A 4×4 array.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of a 4x4 array into two 8-sets

A representative coordinatization:

 

0000  0001  0010  0011
0100  0101  0110  0111
1000  1001  1010  1011
1100  1101  1110  1111

 

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.


Fifteen partitions of an array of 16 triangles into two 8-sets


A representative coordinatization:

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

Saturday, June 16, 2012

Chiral Problem

Filed under: General,Geometry — Tags: , — m759 @ 1:06 AM

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post

"ANDOVERHARVARD THEOLOGICAL LIBRARY"

Detail from Knowles's obituary in this  morning's New York Times

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

From Pilate Goes to Kindergarten

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube

Problem: Is there a different coordinatization
 that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

"Chiral Tetrahedrons as Unitary Quaternions"—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

Monday, February 20, 2012

Coxeter and the Relativity Problem

Filed under: General,Geometry — m759 @ 12:00 PM

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of  the primitive root λ or 10 are

λ = 10,  λ2 = 21,  λ3 = 22,  λ4 = 02,
λ5 = 20,  λ6 = 12,  λ7 = 11,  λ8 = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"

http://www.log24.com/log/pix12/120220-CoxeterFig10.jpg

Coxeter's Figure 10 yields...

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The Aleph

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

Friday, November 25, 2011

Innermost Kernel

Filed under: General — m759 @ 5:01 AM

Thomas Mann on an innermost kernel

http://www.log24.com/log/pix11C/111125-Mann-InnermostKernel.jpg

"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)

See also previous quotations here of the phrase "innermost kernel."

Thursday, November 3, 2011

Ockham’s Bubbles–

Filed under: General,Geometry — m759 @ 10:30 AM

Mathematics and Narrative, continued

"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"

— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th

"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."

— Dan Conover on September 1st, 2011

"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'

I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."

— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996

"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"

Stanford Encyclopedia of Philosophy

"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."

— Interviewer for the National Association of Scholars

From that conversation—

Wolfe : "People in academia should start insisting on objective scholarship, insisting  on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"

As for "mis-imagined quantum pins"…
This 
journal on the date of the above interview— February 28, 2008

http://www.log24.com/log/pix08/080228-Wooters2.jpg

Illustration from a Perimeter Institute talk given on July 20, 2005

The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by  Conover, "The Last Epiphany," and four posts from September 1st, 2011—

BoundaryHow It WorksFor Thor's Day,  and The Galois Tesseract.

Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.

"There is  such a thing as a tesseract." —A Wrinkle in Time

Friday, September 30, 2011

Primordiality

Filed under: General — m759 @ 3:48 PM

"A Phenomenological Perspective,"
Ch. 2 in The Star and the Whole:
Gian-Carlo Rota on Mathematics and Phenomenology 
,
by Fabrizio Palombi, A K Peters/CRC Press, 2011—

"Rota is convinced that one of the fundamental tasks of phenomenology is that of highlighting the primordiality of sense. In his words, if 'many disputes among philosophers are disputes about primordiality' then 'phenomenology is yet another dispute about what is most primordial' (Rota, 1991a,* p. 54). In this way he evidently does not intend to deny the existence of matter, of objects, or of that objective dimension proper to science, in favor of a spiritualist option, but rather to posit as primordial another dimension of the world connected with contexts and with roles, which is considered primordial because each one of us is confronted with it primordially."

* The End of Objectivity: The Legacy of Phenomenology ,
Lectures by Rota at MIT 1974-1991, 457 pages,
MIT Mathematics Department, Cambridge, MA

"The Ultimate, Apocalyptic Laptop"
by George Johnson
Published: September 5, 2000, by The New York Times

"In a paper in the current issue of Nature , Dr. Lloyd describes the ultimate laptop— a computer as powerful as the laws of physics will allow. So energetic is this imaginary machine that using it would be like harnessing a thermonuclear reaction. In the most extreme version of this computer supreme, so much computational circuitry would be packed into so small a space that the whole thing would collapse and form a tiny black hole, an object so dense that not even light can escape its gravity."

Related material: Rota and "Black Hole" in this journal, as well as the Sator Square.

Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 PM

Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity

From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—

"… we are saying much more than that G M 24 is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M 24 is generated by a set
of seven involutions which possesses all the symmetries of L3(2)
acting on the points of the 7-point projective plane…."
Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
Symmetric Generation , p. 42

See also (click to enlarge)—

http://www.log24.com/log/pix11B/110921-CassirerOnObjectivity-400w.jpg

Cassirer's remarks connect the concept of objectivity  with that of object .

The above quotations perhaps indicate how the Mathieu group M 24 may be viewed as an object.

"This is the moment which I call epiphany. First we recognise that the object is one  integral thing, then we recognise that it is an organised composite structure, a thing  in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that  thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."

— James Joyce, Stephen Hero

For a simpler object "which possesses all the symmetries of L3(2) acting on the points of the 7-point projective plane…." see The Eightfold Cube.

For symmetric generation of L3(2) on that cube, see A Simple Reflection Group of Order 168.

Tuesday, September 20, 2011

Relativity Problem Revisited

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 AM

A footnote was added to Finite Relativity

Background:

Weyl on what he calls the relativity problem

IMAGE- Weyl in 1949 on the relativity problem

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

– Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on  coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M 24.  That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols.  His abstract for a 1990 paper says that in his construction "The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters…."

See "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," by R.T. Curtis,  Mathematical Proceedings of the Cambridge Philosophical Society  (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, "Natural Constructions of the Mathieu groups," Math. Proc. Cambridge Philos. Soc.  (1989), Vol. 106, pp. 423-429

R.T. Curtis. "Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12  and M 24" In Proceedings of 1990 LMS Durham Conference 'Groups, Combinatorics and Geometry'  (eds. M. W. Liebeck and J. Saxl),  London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, "A Survey of Symmetric Generation of Sporadic Simple Groups," in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

Saturday, September 17, 2011

The Uploading

Filed under: General — m759 @ 11:32 AM

(Continued from March 9.)

A detail from "Feist Sings 1, 2, 3, 4"—

"Uploaded by SesameStreet on Jul 18, 2008"

Those who prefer, as Weyl put it,
"
the hard core of objectivity"
to, as Eddington put it,
"the colorful tale of the subjective storyteller mind"
may consult this journal on the same day… July 18, 2008.

Saturday, August 20, 2011

Castle Rock

Filed under: General,Geometry — m759 @ 6:29 PM

Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)

"The metaphor for metamorphosis…" —Endgame

Related material:

"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."

Scholarly paper on "Correlative Cosmologies"

"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.

The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….

'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."

— Alastair Macaulay in today's New York Times

SOCRATES: Let us turn off the road a little….

Libretto for Mark Morris's 'Socrates'

See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:

A 2x4 array of squares

The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S8
acting on the above eight-part figure.

See also The Moore Correspondence from last year
on today's date, August 20.

For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—

http://www.log24.com/log/pix11B/110820-Parallelisms60.jpg

"At the still point, there the dance is."
               — Four Quartets

Tuesday, June 21, 2011

Piracy Project

Filed under: General,Geometry — Tags: , — m759 @ 2:02 AM

Recent piracy of my work as part of a London art project suggests the following.

http://www.log24.com/log/pix11A/110620-PirateWithParrotSm.jpg

           From http://www.trussel.com/rls/rlsgb1.htm

The 2011 Long John Silver Award for academic piracy
goes to ….

Hermann Weyl, for the remark on objectivity and invariance
in his classic work Symmetry  that skillfully pirated
the much earlier work of philosopher Ernst Cassirer.

And the 2011 Parrot Award for adept academic idea-lifting
goes to …

Richard Evan Schwartz of Brown University, for his
use, without citation, of Cullinane’s work illustrating
Weyl’s “relativity problem” in a finite-geometry context.

For further details, click on the above names.

Monday, June 20, 2011

The Search for Invariants

Filed under: General,Geometry — m759 @ 9:29 AM

The title of a recent contribution to a London art-related "Piracy Project" begins with the phrase "The Search for Invariants."

A search for that phrase  elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."

Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."

Cassirer's concepts seem related to Weyl's famous remark that

Objectivity means invariance with respect to the group of automorphisms.”
Symmetry  (Princeton University Press, 1952, page 132)

See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.

Update of 6 to 7:50 PM June 20, 2011—

Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.

Another source on Cassirer, invariance, and objectivity

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

A search in Weyl's Symmetry  for any reference to Ernst Cassirer yields no results.

* Published in French in 1938.

Saturday, May 7, 2011

Annals of Mathematics

Filed under: General — m759 @ 10:35 PM

University Diaries praised today the late Robert Nozick's pedagogical showmanship.

His scholarship was less praiseworthy. His 2001 book Invariances: The Structure of the Objective World  failed, quite incredibly, to mention Hermann Weyl's classic summary of  the connection between invariance and objectivity.  See a discussion of Nozick in The New York Review of Books  of December 19, 2002

"… one should mention, first and foremost, the mathematician Hermann Weyl who was almost obsessed by this connection. In his beautiful little book Symmetry  he tersely says, 'Objectivity means invariance with respect to the group of automorphisms….'"

See also this journal on Dec. 3, 2002, and Feb. 20, 2007.

For some context, see a search on the word stem "objectiv-" in this journal.

Friday, January 14, 2011

Ironic Butterfly

Filed under: General,Geometry — Tags: — m759 @ 11:07 AM

David Brooks's column today quotes Niebuhr. From the same source—
Reinhold Niebuhr, The Irony of American History

Chapter 8: The Significance of Irony

Any interpretation of historical patterns and configurations raises the question whether the patterns, which the observer discerns, are "objectively" true or are imposed upon the vast stuff of history by his imagination. History might be likened to the confusion of spots on the cards used by psychiatrists in a Rorschach test. The patient is asked to report what he sees in these spots; and he may claim to find the outlines of an elephant, butterfly or frog. The psychiatrist draws conclusions from these judgments about the state of the patient’s imagination rather than about the actual configuration of spots on the card. Are historical patterns equally subjective?
….
The Biblical view of human nature and destiny moves within the framework of irony with remarkable consistency. Adam and Eve are expelled from the Garden of Eden because the first pair allowed "the serpent" to insinuate that, if only they would defy the limits which God had set even for his most unique creature, man, they would be like God. All subsequent human actions are infected with a pretentious denial of human limits. But the actions of those who are particularly wise or mighty or righteous fall under special condemnation. The builders of the Tower of Babel are scattered by a confusion of tongues because they sought to build a tower which would reach into the heavens.

Niebuhr's ironic butterfly may be seen in the context of last
Tuesday's post Shining and of last Saturday's noon post True Grid

http://www.log24.com/log/pix11/110114-AlderTilleyColored.gif

The "butterfly" in the above picture is a diagram showing the 12 lines* of the Hesse configuration from True Grid.

It is also a reference to James Hillman's classical image (see Shining) of the psyche, or soul, as a butterfly.

Fanciful, yes, but this is in exact accordance with Hillman's remarks on the soul (as opposed to the spirit— see Tuesday evening's post).

The 12-line butterfly figure may be viewed as related to the discussions of archetypes and universals in Hillman's Re-Visioning Psychology  and in Charles Williams's The Place of the Lion . It is a figure intended here to suggest philosophy, not entertainment.

Niebuhr and Williams, if not the more secular Hillman, might agree that those who value entertainment above all else may look forward to a future in Hell (or, if they are lucky, Purgatory). Perhaps such a future might include a medley of Bob Lind's "Elusive Butterfly" and Iron Butterfly's "In-a-Gadda-da-Vida."

* Three horizontal, three vertical, two diagonal, and four arc-shaped.

Saturday, January 8, 2011

True Grid (continued)

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

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

Sunday December 10, 2006

 

 m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
– Thomas Pynchon   

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Sunday, October 17, 2010

An Intricate Reflection

Filed under: General — m759 @ 2:00 AM

"Humanity's fascination with numbers is ancient and complex. Our present relationship with numbers reveals both a highly developed tool and a highly developed user, working together to measure, create, and predict both ourselves and the world around us. But like every symbiotic couple, the tool we would like to believe is separate from us (and thus objective) is actually an intricate reflection of our thoughts, interests, and capabilities."

The Secret Lives of Numbers, by New Radio and Performing Arts

(recommended on the Frivolous Linkages page at Daniel Gilbert's Harvard website)

Other linkages:

New York Lottery on October 16: Midday 706, Evening 684.

Related material — 7/06, 2007, and post no. 684 in this journal.

The above "Secret Lives of Numbers" quotation was suggested by Gilbert's "Magic by Numbers" op-ed piece in today's New York Times

http://www.log24.com/log/pix10B/101017-MagicByNumbers.jpg

"Ay que bonito es volar…"

Wednesday, August 18, 2010

The Sense of an Ending

Filed under: General — Tags: — m759 @ 4:23 PM

Sir Frank Kermode died yesterday (British time) at 90.

“Time cannot exist without a soul (to count it).” — Aristotle

— Passage quoted on the title page of Kermode’s The Sense of an Ending  (Oxford University Press, 1967)

The Cambridge Companion to Plotinus, Lloyd P. Gerson, Cambridge University Press, 1996, p. 208—

“Although Aristotle seems in general to regard time as something independent of the soul and objective, he occasionally gives a leading role to soul. He says, for example, that time cannot exist without a soul to number it (Phys. 223a21-9)….”

http://www.log24.com/log/pix10B/100818-AristotlePhysics223.gif

Soul Riff for Sir Frank— See

  1. An obituary for D-Day piper Bill Millin that says he also died on August 17 (British time)
  2. A Log24 post for the day that Peter O’Toole turned 70
  3. O’Toole in the 1967 Casino Royale.

Wednesday, June 23, 2010

Group Theory and Philosophy

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM

Excerpts from "The Concept of Group and the Theory of Perception,"
by Ernst Cassirer, Philosophy and Phenomenological Research,
Volume V, Number 1, September, 1944.
(Published in French in the Journal de Psychologie, 1938, pp. 368-414.)

The group-theoretical interpretation of the fundaments of geometry is,
from the standpoint of pure logic, of great importance, since it enables us to
state the problem of the "universality" of mathematical concepts in simple
and precise form and thus to disentangle it from the difficulties and ambigui-
ties with which it is beset in its usual formulation. Since the times of the
great controversies about the status of universals in the Middle Ages, logic
and psychology have always been troubled with these ambiguities….

Our foregoing reflections on the concept of group  permit us to define more
precisely what is involved in, and meant by, that "rule" which renders both
geometrical and perceptual concepts universal. The rule may, in simple
and exact terms, be defined as that group of transformations  with regard to
which the variation of the particular image is considered. We have seen
above that this conception operates as the constitutive principle in the con-
struction of the universe of mathematical concepts….

                                                              …Within Euclidean geometry,
a "triangle" is conceived of as a pure geometrical "essence," and this
essence is regarded as invariant with respect to that "principal group" of
spatial transformations to which Euclidean geometry refers, viz., displace-
ments, transformations by similarity. But it must always be possible to
exhibit any particular figure, chosen from this infinite class, as a concrete
and intuitively representable object. Greek mathematics could not
dispense with this requirement which is rooted in a fundamental principle
of Greek philosophy, the principle of the correlatedness of "logos" and
"eidos." It is, however, characteristic of the modern development of
mathematics, that this bond between "logos" and "eidos," which was indis-
soluble for Greek thought, has been loosened more and more, to be, in the
end, completely broken….

                                                            …This process has come to its logical
conclusion and systematic completion in the development of modern group-
theory. Geometrical figures  are no longer regarded as fundamental, as
date of perception or immediate intuition. The "nature" or "essence" of a
figure is defined in terms of the operations  which may be said to
generate the figure.
The operations in question are, in turn, subject to
certain group conditions….

                                                                                                    …What we
find in both cases are invariances with respect to variations undergone by
the primitive elements out of which a form is constructed. The peculiar
kind of "identity" that is attributed to apparently altogether heterogen-
eous figures in virtue of their being transformable into one another by means
of certain operations defining a group, is thus seen to exist also in the
domain of perception. This identity permits us not only to single out ele-
ments but also to grasp "structures" in perception. To the mathematical
concept of "transformability" there corresponds, in the domain of per-
ception, the concept of "transposability." The theory  of the latter con-
cept has been worked out step by step and its development has gone through
various stages….
                                                                                 …By the acceptance of
"form" as a primitive concept, psychological theory has freed it from the
character of contingency  which it possessed for its first founders. The inter-
pretation of perception as a mere mosaic of sensations, a "bundle" of simple
sense-impressions has proved untenable…. 

                             …In the domain of mathematics this state of affairs mani-
fests itself in the impossibility of searching for invariant properties of a
figure except with reference to a group. As long as there existed but one
form of geometry, i.e., as long as Euclidean geometry was considered as the
geometry kat' exochen  this fact was somehow concealed. It was possible
to assume implicitly  the principal group of spatial transformations that lies
at the basis of Euclidean geometry. With the advent of non-Euclidean
geometries, however, it became indispensable to have a complete and sys-
tematic survey of the different "geometries," i.e., the different theories of
invariancy that result from the choice of certain groups of transformation.
This is the task which F. Klein set to himself and which he brought to a
certain logical fulfillment in his Vergleichende Untersuchungen ueber neuere
geometrische Forschungen
….

                                                          …Without discrimination between the
accidental and the substantial, the transitory and the permanent, there
would be no constitution of an objective reality.

This process, unceasingly operative in perception and, so to speak, ex-
pressing the inner dynamics of the latter, seems to have come to final per-
fection, when we go beyond perception to enter into the domain of pure
thought. For the logical advantage and peculiar privilege of the pure con –
cept seems to consist in the replacement of fluctuating perception by some-
thing precise and exactly determined. The pure concept does not lose
itself in the flux of appearances; it tends from "becoming" toward "being,"
from dynamics toward statics. In this achievement philosophers have
ever seen the genuine meaning and value of geometry. When Plato re-
gards geometry as the prerequisite to philosophical knowledge, it is because
geometry alone renders accessible the realm of things eternal; tou gar aei
ontos he geometrike gnosis estin
. Can there be degrees or levels of objec-
tive knowledge in this realm of eternal being, or does not rather knowledge
attain here an absolute maximum? Ancient geometry cannot but answer
in the affirmative to this question. For ancient geometry, in the classical
form it received from Euclid, there was such a maximum, a non plus ultra.
But modern group theory thinking has brought about a remarkable change
In this matter. Group theory is far from challenging the truth of Euclidean
metrical geometry, but it does challenge its claim to definitiveness. Each
geometry is considered as a theory of invariants of a certain group; the
groups themselves may be classified in the order of increasing generality.
The "principal group" of transformations which underlies Euclidean geome-
try permits us to establish a number of properties that are invariant with
respect to the transformations in question. But when we pass from this
"principal group" to another, by including, for example, affinitive and pro-
jective transformations, all that we had established thus far and which,
from the point of view of Euclidean geometry, looked like a definitive result
and a consolidated achievement, becomes fluctuating again. With every
extension of the principal group, some of the properties that we had taken
for invariant are lost. We come to other properties that may be hierar-
chically arranged. Many differences that are considered as essential
within ordinary metrical geometry, may now prove "accidental." With
reference to the new group-principle they appear as "unessential" modifica-
tions….

                 … From the point of view of modern geometrical systematization,
geometrical judgments, however "true" in themselves, are nevertheless not
all of them equally "essential" and necessary. Modern geometry
endeavors to attain progressively to more and more fundamental strata of
spatial determination. The depth of these strata depends upon the com-
prehensiveness of the concept of group; it is proportional to the strictness of
the conditions that must be satisfied by the invariance that is a universal
postulate with respect to geometrical entities. Thus the objective truth
and structure of space cannot be apprehended at a single glance, but have to
be progressively  discovered and established. If geometrical thought is to
achieve this discovery, the conceptual means that it employs must become
more and more universal….

Monday, June 7, 2010

Inspirational Combinatorics

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

According to the Mathematical Association of America this morning, one purpose of the upcoming June/July issue of the Notices of the American Mathematical Society  is

"…to stress the inspirational role of combinatorics…."

Here is another contribution along those lines—

Eidetic Variation

from page 244 of
From Combinatorics to Philosophy: The Legacy of  G.-C. Rota,
hardcover, published by Springer on August 4, 2009

(Edited by Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, and Fabrizio Palombi)

"Rota's Philosophical Insights," by Massimo Mugnai—

"… In other words, 'objectivism' is the attitude [that tries] to render a particular aspect absolute and dominant over the others; it is a kind of narrow-mindedness attempting to reduce to only one the multiple layers which constitute what we call 'reality.' According to Rota, this narrow-mindedness limits in an essential way even of [sic ] the most basic facts of our cognitive activity, as, for example, the understanding of a simple declarative sentence: 'So objectivism is the error we [make when we] persist in believing that we can understand what a declarative sentence means without a possible thematization of this declarative sentence in one of [an] endless variety of possible contexts' (Rota, 1991*, p. 155). Rota here implicitly refers to what, amongst phenomenologists is known as eidetic variation, i.e. the change of perspective, imposed by experience or performed voluntarily, from which to look at things, facts or sentences of the world. A typical example, proposed by Heidegger, in Sein und Zeit  (1927) and repeated many times by Rota, is that of the hammer."

* Rota, G.-C. (1991), The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT, Cambridge, MA, MIT Mathematics Department

The example of the hammer appears also on yesterday's online New York Times  front page—

http://www.log24.com/log/pix10A/100606-Touchstones.jpg

Related material:

From The Blackwell Dictionary of Western Philosophy

Eidetic variation — an alternative expression for eidetic reduction

Eidetic reduction

Husserl's term for an intuitive act toward an essence or universal, in contrast to an empirical intuition or perception. He also called this act an essential intuition, eidetic intuition, or eidetic variation. In Greek, eideo  means “to see” and what is seen is an eidos  (Platonic Form), that is, the common characteristic of a number of entities or regularities in experience. For Plato, eidos  means what is seen by the eye of the soul and is identical with essence. Husserl also called this act “ideation,” for ideo  is synonymous with eideo  and also means “to see” in Greek. Correspondingly, idea  is identical to eidos.

An example of eidos— Plato's diamond (from the Meno )—

http://www.log24.com/log/pix10A/100607-PlatoDiamond.gif

For examples of variation of this eidos, see the diamond theorem.
See also Blockheads (8/22/08).

Related poetic remarks— The Trials of Device.

Saturday, February 27, 2010

Cubist Geometries

Filed under: General,Geometry — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

Oxley's 2004 drawing of the 13-point projective plane

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

Saturday, February 20, 2010

The Mathieu Relativity Problem

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

Weyl on what he calls the relativity problem

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, 1946, The Classical Groups, Princeton University Press, p. 16

Twenty-four years ago a note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M24.  That is, there is no obvious way to apply exactly 24 distinct transformable coordinates (or symbol-strings) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M24.

There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M24.

R.D. Carmichael in 1931 on his construction of the Steiner system S(5,8,24)–

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

— R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Friday, January 22, 2010

Yesterday’s Man

Filed under: General — m759 @ 12:00 PM

Anne Applebaum in the current New York Review of Books on Arthur Koestler

"At the moment, he still seems like yesterday's man, unfashionable and obsolete."

Rather like God. See this journal yesterday– Darkness at Noon.

See also David Levine's portrait of Koestler (Dec. 30, 2009)–

http://www.log24.com/log/pix09A/091230-Koestlerr-NYRB19641217.gif

— and an objective correlative to yesterday's post —

LA mayor says storm front will hit region at noon on June 21, 2010

Click to enlarge.

Sunday, September 27, 2009

Sunday September 27, 2009

Filed under: General,Geometry — m759 @ 3:00 AM
A Pleasantly
Discursive Treatment

In memory of Unitarian
minister Forrest Church,
 dead at 61 on Thursday:

NY Times Sept. 27, 2009, obituaries, featuring Unitarian minister Forrest Church

Unitarian Universalist Origins: Our Historic Faith

“In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history.”

Gravity’s Rainbow–

“For every kind of vampire, there is a kind of cross.”

Unitarian minister Richard Trudeau

“… I called the belief that

(1) Diamonds– informative, certain truths about the world– exist

the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that

(2) The theorems of Euclidean geometry are diamonds….

As the news about non-Euclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.

Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.

Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The Non-Euclidean Revolution, p. 255

H. S. M. Coxeter, 1987, introduction to Trudeau’s book

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…

A kind
 of cross:

Diamond formed by four diagonally-divided two-color squares

See both
Theme and
Variations
and some more
poetic remarks,

Mirror-Play
 of the Fourfold.

* As recent Log24 entries have pointed out, diamond theory (in the original 1976 sense) is a type of non-Euclidean geometry, since finite geometry is not Euclidean geometry– and is, therefore, non-Euclidean, in the strictest sense (though not according to popular usage).

Thursday, August 20, 2009

Thursday August 20, 2009

Filed under: General,Geometry — m759 @ 4:00 PM

Sophists

From David Lavery’s weblog today

Kierkegaard on Sophists:

“If the natural sciences had been developed in Socrates’ day as they are now, all the sophists would have been scientists. One would have hung a microscope outside his shop in order to attract customers, and then would have had a sign painted saying: Learn and see through a giant microscope how a man thinks (and on reading the advertisement Socrates would have said: that is how men who do not think behave).”

— Søren Kierkegaard, Journals, edited and translated by Alexander Dru

To anyone familiar with Pirsig’s classic Zen and the Art of Motorcycle Maintenance, the above remarks of Kierkegaard ring false. Actually, the sophists as described by Pirsig are not at all like scientists, but rather like relativist purveyors of postmodern literary “theory.” According to Pirsig, the scientists are like Plato (and hence Socrates)– defenders of objective truth.

Pirsig on Sophists:

“The pre-Socratic philosophers mentioned so far all sought to establish a universal Immortal Principle in the external world they found around them. Their common effort united them into a group that may be called Cosmologists. They all agreed that such a principle existed but their disagreements as to what it was seemed irresolvable. The followers of Heraclitus insisted the Immortal Principle was change and motion. But Parmenides’ disciple, Zeno, proved through a series of paradoxes that any perception of motion and change is illusory. Reality had to be motionless.

The resolution of the arguments of the Cosmologists came from a new direction entirely, from a group Phædrus seemed to feel were early humanists. They were teachers, but what they sought to teach was not principles, but beliefs of men. Their object was not any single absolute truth, but the improvement of men. All principles, all truths, are relative, they said. ‘Man is the measure of all things.’ These were the famous teachers of ‘wisdom,’ the Sophists of ancient Greece.

To Phaedrus, this backlight from the conflict between the Sophists and the Cosmologists adds an entirely new dimension to the Dialogues of Plato. Socrates is not just expounding noble ideas in a vacuum. He is in the middle of a war between those who think truth is absolute and those who think truth is relative. He is fighting that war with everything he has. The Sophists are the enemy.

Now Plato’s hatred of the Sophists makes sense. He and Socrates are defending the Immortal Principle of the Cosmologists against what they consider to be the decadence of the Sophists. Truth. Knowledge. That which is independent of what anyone thinks about it. The ideal that Socrates died for. The ideal that Greece alone possesses for the first time in the history of the world. It is still a very fragile thing. It can disappear completely. Plato abhors and damns the Sophists without restraint, not because they are low and immoral people… there are obviously much lower and more immoral people in Greece he completely ignores. He damns them because they threaten mankind’s first beginning grasp of the idea of truth. That’s what it is all about.

The results of Socrates’ martyrdom and Plato’s unexcelled prose that followed are nothing less than the whole world of Western man as we know it. If the idea of truth had been allowed to perish unrediscovered by the Renaissance it’s unlikely that we would be much beyond the level of prehistoric man today. The ideas of science and technology and other systematically organized efforts of man are dead-centered on it. It is the nucleus of it all.

And yet, Phaedrus understands, what he is saying about Quality is somehow opposed to all this. It seems to agree much more closely with the Sophists.”

I agree with Plato’s (and Rebecca Goldstein’s) contempt for relativists. Yet Pirsig makes a very important point. It is not the scientists but rather the storytellers (not, mind you, the literary theorists) who sometimes seem to embody Quality.

As for hanging a sign outside the shop, I suggest (particularly to New Zealand’s Cullinane College) that either or both of the following pictures would be more suggestive of Quality than a microscope:

Alfred Bester covers showing 'primordial protomatter' (altered here) from 'Stars' and Rogue Winter from 'Deceivers'

For the “primordial protomatter”
in the picture at left, see
The Diamond Archetype.

Monday, March 16, 2009

Monday March 16, 2009

Filed under: General — m759 @ 8:00 PM
Damnation Morning
continued

Annals of Prose Style

  Film Review

“No offense to either of them, but ‘Georgia Rule’ suggests an Ingmar Bergman script as directed by Jerry Lewis. The subject matter is grim, the relationships are gnarled, the worldview is bleak, and, at any given moment, you suspect someone’s going to be hit with a pie.” –John Anderson at Variety.com, May 8, 2007

Sounds perfect to me.


Through a Glass Darkly

“Preserving a strict unity of time and place, this stark tale of a young woman’s decline into insanity is set in a summer home on a holiday island. It is the first part of the trilogy that comprises Winter Light and The Silence, films which are generally seen as addressing Bergman’s increasing disillusionment with the emotional coldness of his inherited Lutheran religion. In particular here, Bergman focuses on the absence of familial love which might perhaps have pulled Karin (Andersson) back from the brink; while Karin’s mental disintegration manifests itself in the belief that God is a spider. As she slips inexorably into madness, she is observed with terrifying objectivity by her emotionally paralyzed father (Björnstrand) and seemingly helpless husband (von Sydow).”

— Nigel Floyd, Time Out, quoted at Bergmanorama

Related material:

1. The “spider” symbol of Fritz Leiber’s short story “Damnation Morning”–

2. Hollywood’s “Angels & Demons” (to open May 15), and

3. The following diagram by one “John Opsopaus”–

http://www.log24.com/log/pix09/090312-OpsopausSquare.jpg

Monday March 16, 2009

Filed under: General — m759 @ 8:00 PM
Damnation Morning
continued

Annals of Prose Style

  Film Review

“No offense to either of them, but ‘Georgia Rule’ suggests an Ingmar Bergman script as directed by Jerry Lewis. The subject matter is grim, the relationships are gnarled, the worldview is bleak, and, at any given moment, you suspect someone’s going to be hit with a pie.” –John Anderson at Variety.com, May 8, 2007

Sounds perfect to me.


Through a Glass Darkly

“Preserving a strict unity of time and place, this stark tale of a young woman’s decline into insanity is set in a summer home on a holiday island. It is the first part of the trilogy

Bergman's trilogy including 'Through a Glass Darkly'

that comprises Winter Light and The Silence, films which are generally seen as addressing Bergman’s increasing disillusionment with the emotional coldness of his inherited Lutheran religion. In particular here, Bergman focuses on the absence of familial love which might perhaps have pulled Karin (Andersson) back from the brink; while Karin’s mental disintegration manifests itself in the belief that God is a spider. As she slips inexorably into madness, she is observed with terrifying objectivity by her emotionally paralyzed father (Björnstrand) and seemingly helpless husband (von Sydow).”

— Nigel Floyd, Time Out, quoted at Bergmanorama

Related material:

1. The “spider” symbol of Fritz Leiber’s short story “Damnation Morning“–

Fritz Leiber's 'spider' figure

2. The Illuminati Diamond of Hollywood’s “Angels & Demons” (to open May 15), and

3. The following diagram by one “John Opsopaus“–

Elemental square by John Opsopaus from 'The Rotation of the Elements'

Friday, February 20, 2009

Friday February 20, 2009

Filed under: General,Geometry — Tags: , — m759 @ 2:01 PM
Emblematizing
 the Modern
 

The following meditation was
inspired by the recent fictional
recovery, by Mira Sorvino
in "The Last Templar,"

of a Greek Cross —
"the Cross of Constantine"–
and by the discovery, by
art historian Rosalind Krauss,
of a Greek Cross in the
art of Ad Reinhardt.

http://www.log24.com/log/pix09/090220-CrossOfDescartes.jpg

The Cross of Descartes  

Note that in applications, the vertical axis
of the Cross of Descartes often symbolizes
the timeless (money, temperature, etc.)
while the horizontal axis often symbolizes time.


T.S. Eliot:

"Men’s curiosity searches past and future
And clings to that dimension. But to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint…."


There is a reason, apart from her ethnic origins, that Rosalind Krauss (cf. 9/13/06) rejects, with a shudder, the cross as a key to "the Pandora's box of spiritual reference that is opened once one uses it." The rejection occurs in the context of her attempt to establish not the cross, but the grid, as a religious symbol:
 

"In suggesting that the success [1] of the grid
is somehow connected to its structure as myth,
I may of course be accused of stretching a point
beyond the limits of common sense, since myths
are stories, and like all narratives they unravel
through time, whereas grids are not only spatial
to start with, they are visual structures
that explicitly reject a narrative
or sequential reading of any kind.

[1] Success here refers to
three things at once:
a sheerly quantitative success,
involving the number of artists
in this century who have used grids;
a qualitative success through which
the grid has become the medium
for some of the greatest works
of modernism; and an ideological
success, in that the grid is able–
in a work of whatever quality–
to emblematize the Modern."

— Rosalind Krauss, "Grids" (1979)

Related material:

Time Fold and Weyl on
objectivity and frames of reference.

See also Stambaugh on
The Formless Self
as well as
A Study in Art Education
and
Jung and the Imago Dei.

Thursday, February 19, 2009

Thursday February 19, 2009

Filed under: General — m759 @ 7:07 AM

A Sunrise
for Sunrise

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete."

Rosalind Krauss, "Grids"

Yesterday's entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo)

The 3x3 grid

may be used to create a graphic design– a Greek cross, which appears also in crossword puzzles:

Crossword-puzzle design that includes Greek-cross elements

Illustration by
Paul Rand
(born Peretz Rosenbaum)

A more sophisticated example
of the ninefold square
in graphic design:

"That old Jew
gave me this here."

— A Flag for Sunrise  

The 3x3 grid as an organizing frame for Chinese calligraphy. Example-- the character for 'sunrise'
From Paul-Rand.com

Friday, December 5, 2008

Friday December 5, 2008

Filed under: General — m759 @ 4:30 PM
Continued from Monday:

A Version of
Heaven’s Gate

in memory of
Alexy II, the Russian Orthodox
 patriarch who died today in Moscow:

Art logo: frame not X'd out

The Pearly Gates of Cyberspace:

From Geoffrey Broadbent,
“Why a Black Square?” in Malevich
 (London, Art and Design/
Academy Group, 1989, p. 49):

Malevich’s Black Square seems to be
nothing more, nor less, than his
‘Non-Objective’ representation
of Bragdon’s (human-being-as) Cube
  passing through the ‘Plane of Reality.’!”

Sunday, August 24, 2008

Sunday August 24, 2008

Filed under: General — m759 @ 7:00 AM
Cross-Purposes

Yesterday’s entry, Absurdities, quoted Erich Heller:

“All relevant objective truths are born and die as absurdities. They come into being as the monstrous claim of an inspired rebel and pass away with the eccentricity of a superstitious crank.”

The context for this remarkable saying is Heller’s essay “The Hazard of Modern Poetry.” (See p. 270 in the links below.)

Discussing “the century of Pascal and Hobbes,” he says (see the link to p. 269 below) that

“… as for spiritual cunning, it was in the conceits of metaphysical poetry, in the self-conscious ambiguities of poetical language (there are, we are told, as many types of it as deadly sins), and in the paradoxes of Pascal’s religious thought. For ambiguity and paradox are the manner of speaking when reality and symbol, man’s mind and his soul, are at cross-purposes.”

Heller’s description of “relevant objective truths” as “absurdities” seems to be an instance of such ambiguity and paradox. For further details, see

The Disinherited Mind: Essays in Modern German Literature and Thought (Harvest paperback, 1975)–

“The Hazard of Modern Poetry” (pp. 263-300), Section 1, pages

263, 264, 265, 266, 267, 268, 269, 270, 271, 272.

For material related to Pascal, see the five Log24 entries ending on D-Day, 2008.

For material related to Hobbes, see the five Log24 entries ending on St. Patrick’s Day, 2007.

Saturday, August 23, 2008

Saturday August 23, 2008

Filed under: General — m759 @ 5:01 AM
Absurdities

“The balance-beam of Fate was bent;  
The bounds of good and ill were rent;  
Strong Hades could not keep his own,  
But all slid to confusion.”

— “Uriel,” by  
Ralph Waldo Emerson:

Oxford Book of
English Verse
, 1919,
number
 670

“All relevant objective truths are born and die as absurdities. They come into being as the monstrous claim of an inspired rebel and pass away with the eccentricity of a superstitious crank.”

— Erich Heller, The Disinherited Mind

NY lottery Aug. 22, 2008: mid-day 670, evening 666

Related material:

Yesterday’s entry
and
Angels in Arabia

Saturday, July 19, 2008

Saturday July 19, 2008

Filed under: General,Geometry — m759 @ 2:00 PM
Hard Core

(continued from yesterday)

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week’s New Yorker:

A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent ‘object’ in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure.”

Hermann Weyl on the hard core of objectivity:

“Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind.” (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)


Steven H. Cullinane on the symmetries of a 4×4 array of points:

A Structure-Endowed Entity

“A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way.”

— Hermann Weyl in Symmetry

Let us apply Weyl’s lesson to the following “structure-endowed entity.”

4x4 array of dots

What is the order of the resulting group of automorphisms?

The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
  called a “colorful tale”–

The Diamond 16 Puzzle

The Diamond 16 Puzzle

This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.

“You can make 322,560
pairs of patterns. Each
 pair pictures a different
symmetry of the underlying
16-point space.”

— Steven H. Cullinane,
July 17, 2008

For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings
.

Friday, July 18, 2008

Friday July 18, 2008

Filed under: General,Geometry — m759 @ 12:00 PM

Hard Core

David Corfield quotes Weyl in a weblog entry, "Hierarchy and Emergence," at the n-Category Cafe this morning:

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science [Princeton, 1949], p. 237)

For the same quotation in a combinatorial context, see the foreword by A. W. Tucker, "Combinatorial Problems," to a special issue of the IBM Journal of Research and Development, November 1960 (1-page pdf).

See also yesterday's Log24 entry.
 

Tuesday, July 8, 2008

Tuesday July 8, 2008

Filed under: General — Tags: — m759 @ 3:33 AM
Translation
 

Yesterday's entry discussed T.E. Hulme— a co-founder, with Ezra Pound, of the Imagist school of poetry. Recent entries on randomness, using the New York Lottery as a source of examples, together with Hulme's approach to poetry discussed yesterday, suggest the following meditation– what Charles Cameron might call a "bead game."

Part I:

Ezra Pound on Imagism (from Gaudier-Brzeska,* 1916):

Three years ago in Paris I got out of a "metro" train at La Concorde, and saw suddenly a beautiful face, and then another and another, and then a beautiful child’s face, and then another beautiful woman, and I tried all that day to find words for what this had meant to me, and I could not find any words that seemed to me worthy, or as lovely as that sudden emotion. [….]

The "one image poem" is a form of super-position, that is to say, it is one idea set on top of another. I found it useful in getting out of the impasse in which I had been left by my metro emotion. I wrote a thirty-line poem, and destroyed it because it was what we call work "of second intensity." Six months later I made a poem half that length; a year later I made the following hokku-like sentence: —

"The apparition of these
    faces in the crowd:
 Petals, on a
    wet, black bough."

 

I dare say it is meaningless unless one has drifted into a certain vein of thought. In a poem of this sort one is trying to record the precise instant when a thing outward and objective transforms itself, or darts into a thing inward and subjective.

Part II:

Eleanor Goodman on translation (in a July 7, 2008, weblog entry, "Pound and Process: An Introduction"):

"… all translations exist on an axis. Indeed, they exist in a manifold of many axes intersecting. One axis is that of foreignness and familiarity. One axis is that of structural mimicry, another of melodic mimicry. And one axis is that of semantic fidelity."

Goodman's use of the word "manifold" here is of course poetic, not mathematical.

Part III:

New York Lottery, mid-day on July 7, 2008: 771.

Part IV:

A Google search on manifold 771 reveals that 771 is, according to Google's scanners, an alternate form (a "translation," via structural mimicry) of a script version of the letter M. (See Part V below.)

Part V:

Long version of a 
one-image poem —

"Random apparition:
  manifold translated."

This poem summarizes the
relationship (See Part IV above) of
the (apparently) random number 771
to the rather non-random concept of
a linear manifold:

Paul R. Halmos, Finite Dimensional Vector Spaces, Princeton, 1948-- Definition of linear manifold (denoted by script M)

[Such lines and planes have not
been, in mathematical language,
"translated."]

— Paul R. Halmos,
Finite Dimensional Vector Spaces,
Princeton University Press, 1948

Short version of the   
above one-image poem

 771:
Script M

* Gaudier-Brzeska created the artifact shown on the cover of Solid Objects, a work of literary theory by Douglas Mao. For more on that artifact and on the New York Lottery, see Sermon for St. Peter's Day. "It is not in the premise that reality/ Is a solid…." –Wallace Stevens

"I was like, Oh My God." —Poet Billy Collins at Chautauqua Institution, morning of July 7, 2008
 

Sunday, June 22, 2008

Sunday June 22, 2008

Filed under: General — m759 @ 7:00 AM
Salvation by Grace

Today’s New York Times has an obituary of Henry Chadwick, an Anglican priest and expert on church history who believed strongly in ecumenism.

Church history and ecumenism may interest few Americans, who have not recently suffered the sort of conflicts familiar to Northern Ireland.

Nevertheless, here are some thoughts on the matter.

From a statement of “the five points of Calvinism”–

Irresistible Grace

“‘Irresistible grace’ refers to the grace of regeneration by which God effectually calls His elect inwardly, converting them to Himself, and quickening them from spiritual death to spiritual life.  Regeneration is the sovereign and immediate work of the Holy Spirit….”

Calvinism is, of course, a deeply serious and powerful approach to spiritual matters.

(See 6/3/08 and 2/20/05.)

Still, I prefer the following visions of grace:

How does one stand
To behold the sublime,
To confront the mockers,
The mickey mockers
And plated pairs?

— Wallace Stevens, 1936

Philadelphia stories: Catholic and Protestant versions, starring Grace Kelly and Katharine Hepburn

On the left, a Catholic answer.
On the right, a Protestant answer.

For further details, see 10/16/05.

The above two
Philadelphia stories
have met in a different
vision of Grace:

Grace Kelly and James Stewart in 'Rear Window'

Click image for a (much) larger version.

This tableau, in the larger version showing details in the background buildings, seems to me an apt, if more Calvinist and less Catholic, version of what Paul Simon, in his Graceland album, has memorably called “angels in the architecture.”

Let us hope that the late Henry Chadwick now has a place among such angels.

Related material:

Yesterday’s entries and
what T. S. Eliot might call
their “objective correlatives
in the Pennsylvania Lottery
and in this journal:

PA Lottery Saturday, June 21, 2008: Mid-day 529, Evening 501

5/29

5/01

Sunday, April 13, 2008

Sunday April 13, 2008

Filed under: General,Geometry — m759 @ 7:59 AM
The Echo
in Plato’s Cave

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy.”

— Simon Blackburn, Think (Oxford, 1999)

Michael Harris, mathematician at the University of Paris:

“… three ‘parts’ of tragedy identified by Aristotle that transpose to fiction of all types– plot (mythos), character (ethos), and ‘thought’ (dianoia)….”

— paper (pdf) to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.

Mythos —

A visitor from France this morning viewed the entry of Jan. 23, 2006: “In Defense of Hilbert (On His Birthday).” That entry concerns a remark of Michael Harris.

A check of Harris’s website reveals a new article:

“Do Androids Prove Theorems in Their Sleep?” (slighly longer version of article to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.) (pdf).

From that article:

“The word ‘key’ functions here to structure the reading of the article, to draw the reader’s attention initially to the element of the proof the author considers most important. Compare E.M. Forster in Aspects of the Novel:

[plot is] something which is measured not be minutes or hours, but by intensity, so that when we look at our past it does not stretch back evenly but piles up into a few notable pinnacles.”

Ethos —

“Forster took pains to widen and deepen the enigmatic character of his novel, to make it a puzzle insoluble within its own terms, or without. Early drafts of A Passage to India reveal a number of false starts. Forster repeatedly revised drafts of chapters thirteen through sixteen, which comprise the crux of the novel, the visit to the Marabar Caves. When he began writing the novel, his intention was to make the cave scene central and significant, but he did not yet know how:

When I began a A Passage to India, I knew something important happened in the Malabar (sic) Caves, and that it would have a central place in the novel– but I didn’t know what it would be… The Malabar Caves represented an area in which concentration can take place. They were to engender an event like an egg.”

E. M. Forster: A Passage to India, by Betty Jay

Dianoia —

Flagrant Triviality
or Resplendent Trinity?

“Despite the flagrant triviality of the proof… this result is the key point in the paper.”

— Michael Harris, op. cit., quoting a mathematical paper

Online Etymology Dictionary
:

flagrant
c.1500, “resplendent,” from L. flagrantem (nom. flagrans) “burning,” prp. of flagrare “to burn,” from L. root *flag-, corresponding to PIE *bhleg (cf. Gk. phlegein “to burn, scorch,” O.E. blæc “black”). Sense of “glaringly offensive” first recorded 1706, probably from common legalese phrase in flagrante delicto “red-handed,” lit. “with the crime still blazing.”

A related use of “resplendent”– applied to a Trinity, not a triviality– appears in the Liturgy of Malabar:

http://www.log24.com/log/pix08/080413-LiturgyOfMalabar.jpg

The Liturgies of SS. Mark, James, Clement, Chrysostom, and Basil, and the Church of Malabar, by the Rev. J.M. Neale and the Rev. R.F. Littledale, reprinted by Gorgias Press, 2002

On Universals and
A Passage to India:

“”The universe, then, is less intimation than cipher: a mask rather than a revelation in the romantic sense. Does love meet with love? Do we receive but what we give? The answer is surely a paradox, the paradox that there are Platonic universals beyond, but that the glass is too dark to see them. Is there a light beyond the glass, or is it a mirror only to the self? The Platonic cave is even darker than Plato made it, for it introduces the echo, and so leaves us back in the world of men, which does not carry total meaning, is just a story of events.”

— Betty Jay,  op. cit.

http://www.log24.com/log/pix08/080413-Marabar.jpg

Judy Davis in the Marabar Caves

In mathematics
(as opposed to narrative),
somewhere between
 a flagrant triviality and
a resplendent Trinity we
have what might be called
“a resplendent triviality.”

For further details, see
A Four-Color Theorem.”

Wednesday, October 3, 2007

Wednesday October 3, 2007

Filed under: General,Geometry — m759 @ 3:09 PM
Janitor Monitor

The image “http://www.log24.com/log/pix07A/070803-Trees.jpg” cannot be displayed, because it contains errors.

Will Hunting may be
interested in the following
vacant editorships at
The Open Directory:

Graph Theory
and
Combinatorics.

Related material:

The Long Hello and
On the Holy Trinity

Hey, Carrie-Anne, what’s
your game now….?

The image “http://www.log24.com/log/pix07A/071003-Magdalene.GIF” cannot be displayed, because it contains errors.

Picture sources:
azstarnet.com,
vibrationdata.com.

Personally, I prefer
Carol Ann:

From Criticism,  Fall, 2001,
by Carol Ann Johnston

“Drawing upon Platonic thought, Augustine argues that ideas are actually God’s objective pattern and as such exist in God’s mind. These ideas appear in the mirror of the soul. (35).”

(35.) In Augustine, De Trinitate, trans., Stephen McKenna (Washington, D.C.: Catholic University Press, 1970). See A. B. Acton, “Idealism,” in The Encyclopedia of Philosophy, ed., Paul Edwards. Vol. 4 (New York: Macmillan, 1967): 110-118; Robert McRae, “`Idea’ as a Philosophical Term in the Seventeenth Century,” JHI 26 (1965): 175-190, and Erwin Panofsky, Idea: A Concept in Art History, trans., Joseph J. S. Peake (Columbia, S.C.: University of South Carolina Press, 1968) for explications of this term.

See also
Art Wars: Geometry as Conceptual Art
and Ideas and Art: Notes on Iconology.

For more on Augustine and geometry,
see Today’s Sinner (Aug. 28, 2006).

Tuesday, July 3, 2007

Tuesday July 3, 2007

Filed under: General — m759 @ 9:29 PM
The Ignorance
of Stanley Fish

(continued from
June 18, 2002)

The “ignorance” referred to
is Fish’s ignorance of the
philosophical background
of the words
“particular” and “universal.”

Postmodern Warfare:
The Ignorance of Our
Warrior Intellectuals,”
by Stanley Fish,
Harper’s Magazine,
July 2002, contains
the following passages:

“The deepest strain in a religion is the particular and particularistic doctrine it asserts at its heart, in the company of such pronouncements as ‘Thou shalt have no other Gods before me.’ Take the deepest strain of religion away… and what remains are the surface pieties– abstractions without substantive bite– to which everyone will assent because they are empty, insipid, and safe. It is this same preference for the vacuously general over the disturbingly particular that informs the attacks on college and university professors who spoke out in ways that led them to be branded as outcasts by those who were patrolling and monitoring the narrow boundaries of acceptable speech. Here one must be careful, for there are fools and knaves on all sides.”

“Although it may not at first be obvious, the substitution for real religions of a religion drained of particulars is of a piece with the desire to exorcise postmodernism.”

“What must be protected, then, is the general, the possibility of making pronouncements from a perspective at once detached from and superior to the sectarian perspectives of particular national interests, ethnic concerns, and religious obligations; and the threat to the general is posed by postmodernism and strong religiosity alike, postmodernism because its critique of master narratives deprives us of a mechanism for determining which of two or more fiercely held beliefs is true (which is not to deny the category of true belief, just the possibility of identifying it uncontroversially), strong religiosity because it insists on its own norms and refuses correction from the outside. The antidote to both is the separation of the private from the public, the establishing of a public sphere to which all could have recourse and to the judgments of which all, who are not criminal or insane, would assent. The point of the public sphere is obvious: it is supposed to be the location of those standards and measures that belong to no one but apply to everyone. It is to be the location of the universal. The problem is not that there is no universal–the universal, the absolutely true, exists, and I know what it is. The problem is that you know, too, and that we know different things, which puts us right back where we were a few sentences ago, armed with universal judgments that are irreconcilable, all dressed up and nowhere to go for an authoritative adjudication.

What to do? Well, you do the only thing you can do, the only honest thing: you assert that your universal is the true one, even though your adversaries clearly do not accept it, and you do not attribute their recalcitrance to insanity or mere criminality–the desired public categories of condemnation–but to the fact, regrettable as it may be, that they are in the grip of a set of beliefs that is false. And there you have to leave it, because the next step, the step of proving the falseness of their beliefs to everyone, including those in their grip, is not a step available to us as finite situated human beings. We have to live with the knowledge of two things: that we are absolutely right and that there is no generally accepted measure by which our rightness can be independently validated. That’s just the way it is, and we should just get on with it, acting in accordance with our true beliefs (what else could we do?) without expecting that some God will descend, like the duck in the old Groucho Marx TV show, and tell us that we have uttered the true and secret word.”

From the public spheres
of the Pennsylvania Lottery:

PA Lottery logo

PA Lottery July 3, 2007: Mid-day 105, Evening 268

105 —

Log24 on 1/05:

“‘From your lips
to God’s ears,’
 goes the old
Yiddish wish.

 The writer, by contrast,
tries to read God’s lips
and pass along
the words….”

— Richard Powers   

268 —

This is a page number
that appears, notably,
in my June 2002
journal entry on Fish
,
and again in an entry,
The Transcendent Signified,”
dated July 26, 2003,
that argues against
Fish’s school, postmodernism,
 and in favor of what the pomos
call “logocentrism.”

Page 268
 
of Simon Blackburn’s Think
(Oxford Univ. Press, 1999):

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato’s (realist) reaction to the sophists (nominalists). What is often called ‘postmodernism’ is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth.”

Fish may, if he wishes,
regard the particular
page number 268 as
delivered– five years late,
but such is philosophy–
by Groucho’s
winged messenger
in response to
Fish’s utterance of the
  “true and secret word”–
namely, “universal.”

When not arguing politics,
Fish, though from
a Jewish background, is
 said to be a Milton scholar.
Let us therefore hope he
is by now, or comes to be,
aware of the Christian
approach to universals–
an approach true to the
philosophical background
sketched in 1999 by
Blackburn and made
particular in a 1931 novel
 by Charles Williams,
The Place of the Lion.

Wednesday, March 21, 2007

Wednesday March 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:18 PM
Finite Relativity
continued

This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:

"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives.  If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set  (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant.  For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above.  One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560.  Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."

The epigraph to "Finite Relativity" is:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

The added paragraph seems to fit this description.

Monday, March 19, 2007

Monday March 19, 2007

Filed under: General — Tags: — m759 @ 10:31 AM
The Naked Brain

The cover (pdf) of the Notices of the American Mathematical Society for April 2007 (Mathematics Awareness Month) features a naked disembodied brain (Log24, March 16), courtesy of researchers at the Catholic University of Louvain.
 

Related material:

 

Log24, Jan. 26

"… at last she realized
what the Thing on the dais was.
IT was a brain.
A disembodied brain…."
 
A Wrinkle in Time, by Madeleine L'Engle
"There could not be an objective test
that distinguished a clever robot
from a really conscious person."
 
— Daniel Dennett in TIME magazine,
Daniel Dennett in his office

Daniel Dennett, Professor of Philosophy
and Director of the
Center for Cognitive Studies
at Tufts University,
in his office on campus.
(Boston Globe, Jan. 29, 2006.
Photo © Rick Friedman.)

 

Related recommended
reading and viewing:

Tom Wolfe's essay
"Sorry, But Your Soul Just Died,"
and a video of an interview
 with Wolfe.
 

Tuesday, February 20, 2007

Tuesday February 20, 2007

Filed under: General,Geometry — m759 @ 7:09 AM
Symmetry

Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”

Some relevant quotations:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Describing the branch of mathematics known as Galois theory, Weyl says that it

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

Weyl’s set Sigma is a finite set of complex numbers.   Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes.  For illustrations, see Finite Geometry of the Square and Cube.  What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations.  For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry  Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:

“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].‘[22]

22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).

References:

Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.

Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]

Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.

Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.

See also

Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–

Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–

“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”

References:

Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.

Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].

Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press.  See Invariances: The Structure of the Objective World, by Robert Nozick.

Friday, January 26, 2007

Friday January 26, 2007

Filed under: General — Tags: — m759 @ 2:48 PM
 
IT
 
"… at last she realized
what the Thing on the dais was.
IT was a brain.
A disembodied brain…."
 
A Wrinkle in Time, by Madeleine L'Engle

"There could not be an objective test
that distinguished a clever robot
from a really conscious person."

— Daniel Dennett in TIME magazine,
issue dated Mon., Jan. 29, 2007

 

Daniel Dennett in his office

 

Daniel Dennett, Professor of Philosophy
and Director of the
Center for Cognitive Studies
at Tufts University,
in his office on campus.
(Boston Globe, Jan. 29, 2006.
Photo © Rick Friedman.)

Hexagram 39:
Obstruction

I Ching, Hexagram 39

The Judgment

Obstruction. The southwest furthers.
(See Zenna Henderson.) 
The northeast does not further.
 (See Daniel Dennett.)
It furthers one to see the great man.
 (See Alan Turing.)
Perseverance brings good fortune.

"If telepathy is admitted
it will be necessary
to tighten our test up."
 
Alan Turing, 1950
 
Amen.

Sunday, November 26, 2006

Sunday November 26, 2006

Filed under: General — m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
The Club Dumas,1993

"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
Robert Plant,1971

The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.

An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect    

For more on the field
of reason, see
Log24, Oct. 9, 2006.

A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).

Unreason is, of course,
more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon   
 

Saturday, October 14, 2006

Saturday October 14, 2006

Filed under: General — m759 @ 7:00 PM
The Line
 
Continued
from Aug. 15, 2004:

Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance, Part III:

“The wave of crystallization rolled ahead. He was seeing two worlds, simultaneously. On the intellectual side, the square side, he saw now that Quality was a cleavage term. What every intellectual analyst looks for. You take your analytic knife, put the point directly on the term Quality and just tap, not hard, gently, and the whole world splits, cleaves, right in two…

The Line,
by S. H. Cullinane

hip and square, classic and romantic, technological and humanistic…and the split is clean. There’s no mess. No slop. No little items that could be one way or the other. Not just a skilled break but a very lucky break. Sometimes the best analysts, working with the most obvious lines of cleavage, can tap and get nothing but a pile of trash. And yet here was Quality; a tiny, almost unnoticeable fault line; a line of illogic in our concept of the universe; and you tapped it, and the whole universe came apart, so neatly it was almost unbelievable. He wished Kant were alive. Kant would have appreciated it. That master diamond cutter. He would see. Hold Quality undefined. That was the secret.”

The image “http://www.log24.com/log/pix06A/061014-Kant.gif” cannot be displayed, because it contains errors.

See also the discussion of
subjective and objective
by Robert M. Pirsig in
Zen and the Art of
Motorcycle Maintenance
,
Part III,
followed by this dialogue:

Are We There Yet?

Chris shouts, “When are we
going to get to the top?”

“Probably quite a way yet,”
I reply.

“Will we see a lot?”

“I think so. Look for blue sky
between the trees. As long as we
can’t see sky we know it’s a way yet.
The light will come through the trees
when we round the top.”

Related material:

The Boys from Uruguay,
Lichtung!,
The Shining of May 29,
A Guiding Philosophy,
Ticket Home.

The philosophy of Heidegger
discussed and illustrated
in the above entries may
be regarded as honoring
today’s 100th anniversary
of the birth of Heidegger’s
girlfriend, Hannah Arendt.

See also

 Hannah and Martin
and
Snowblind.

Wednesday, September 13, 2006

Wednesday September 13, 2006

Filed under: General — m759 @ 9:28 PM

ART WARS continued:

The Krauss Cross

The image “http://www.log24.com/log/pix06A/060913-Art.jpg” cannot be displayed, because it contains errors.

Rosalind Krauss in "Grids":

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example, we could think about Ad Reinhardt who, despite his repeated insistence that 'Art is art,' ended up by painting a series of black nine-square grids in which the motif that inescapably emerges is a Greek cross.  There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."

Rebecca Goldstein on
Mathematics and Narrative
:

"I don't write exclusively on Jewish themes or about Jewish characters. My collection of short stories, Strange Attractors, contained nine pieces, five of which were, to some degree, Jewish, and this ratio has provided me with a precise mathematical answer (for me, still the best kind of answer) to the question of whether I am a Jewish writer. I am five-ninths a Jewish writer."

Jacques Maritain,
October 1941
:

"The passion of Israel
today is taking on
more and more distinctly
the form of the Cross."

E. L. Doctorow,
City of God:

"In the garden of Adding,
Live Even and Odd."

Thursday, December 8, 2005

Thursday December 8, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM
Aion Flux

That Nature is a Heraclitean Fire…
— Poem title, Gerard Manley Hopkins  

From Jung’s Map of the Soul, by Murray Stein:

“… Jung thinks of the self as undergoing continual transformation during the course of a lifetime…. At the end of his late work Aion, Jung presents a diagram to illustrate the dynamic movements of the self….”

The image “http://www.log24.com/theory/images/JungDiamonds.gif” cannot be displayed, because it contains errors.

“The formula presents a symbol of the self, for the self is not just a stable quantity or constant form, but is also a dynamic process.  In the same way, the ancients saw the imago Dei in man not as a mere imprint, as a sort of lifeless, stereotyped impression, but as an active force…. The four transformations represent a process of restoration or rejuvenation taking place, as it were, inside the self….”

“The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity. 

What the formula can only hint at, however, is the higher plane that is reached through the process of transformation and integration. The ‘sublimation’ or progress or qualitative change consists in an unfolding of totality into four parts four times, which means nothing less than its becoming conscious. When psychic contents are split up into four aspects, it means that they have been subjected to discrimination by the four orienting functions of consciousness. Only the production of these four aspects makes a total description possible. The process depicted by our formula changes the originally unconscious totality into a conscious one.” 

— Jung, Collected Works, Vol. 9, Part 2, Aion: Researches into the Phenomenology of the Self (1951) 

Related material: 

  The diamond theorem

“Although ‘wholeness’ seems at first sight to be nothing but an abstract idea (like anima and animus), it is nevertheless empirical in so far as it is anticipated by the psyche in the form of  spontaneous or autonomous symbols. These are the quaternity or mandala symbols, which occur not only in the dreams of modern people who have never heard of them, but are widely disseminated in the historical recods of many peoples and many epochs. Their significance as symbols of unity and totality is amply confirmed by history as well as by empirical psychology.  What at first looks like an abstract idea stands in reality for something that exists and can be experienced, that demonstrates its a priori presence spontaneously. Wholeness is thus an objective factor that confronts the subject independently of him… Unity and totality stand at the highest point on the scale of objective values because their symbols can no longer be distinguished from the imago Dei. Hence all statements about the God-image apply also to the empirical symbols of totality.”

— Jung, Aion, as quoted in
Carl Jung and Thomas Merton

Tuesday, November 29, 2005

Tuesday November 29, 2005

Filed under: General — m759 @ 12:25 AM
The Way of the Pilgrim,
Part III:
 
For the Birthday
of C. S. Lewis

The image “http://www.log24.com/log/pix05B/051129-Tao.jpg” cannot be displayed, because it contains errors.

The Tao, Chapter I

“The Chinese… speak of a great thing (the greatest thing) called the Tao. It is the reality beyond all predicates, the abyss that was before the Creator Himself. It is Nature, it is the Way, the Road. It is the Way in which the universe goes on, the Way in which things everlastingly emerge, stilly and tranquilly, into space and time. It is also the Way which every man should tread in imitation of that cosmic and supercosmic progression, conforming all activities to that great exemplar.”

— C. S. Lewis in The Abolition of Man

“In his preface to That Hideous Strength, Lewis says the novel has a serious point that he has tried to make in this little book, The Abolition of Man.  The novel is a work of fantasy or science fiction, while Abolition is a short philosophical work about moral education, but as we shall see the two go together; we will understand either book better by having read and thought about the other.”

— Dale Nelson, Notes on The Abolition of Man

“In Epiphany Term, 1942, C.S. Lewis delivered the Riddell Memorial Lectures… in….  the University of Durham….  He delivered three lectures entitled ‘Men without Chests,’ ‘The Way,’ and ‘The Abolition of Man.’  In them he set out to attack and confute what he saw as the errors of his age. He started by quoting some fashionable lunacy from an educationalists’ textbook, from which he developed a general attack on moral subjectivism.  In his second lecture he argued against various contemporary isms, which purported to replace traditional objective morality.  His final lecture, ‘The Abolition of Man,’ which also provided the title of the book published the following year, was a sustained attack on hard-line scientific anti-humanism. The intervening fifty years have largely vindicated Lewis.”

— J. R. Lucas, The Restoration of Man

Thursday, November 3, 2005

Thursday November 3, 2005

Filed under: General — m759 @ 11:07 AM

Bond

USA Today on last night’s White House dinner:

“In his toast, Bush said the royal visit was ‘a reminder of the unique and enduring bond’ between the two countries.”

From Log24, July 18, 2003:

The use of the word “idea” in my entries’ headlines yesterday was not accidental.  It is related to an occurrence of the word in Understanding: On Death and Truth, a set of journal entries from May 9-12.  The relevant passage on “ideas” is quoted there, within commentary by an Oberlin professor:

“That the truth we understand must be a truth we stand under is brought out nicely in C. S. Lewis’ That Hideous Strength when Mark Studdock gradually learns what an ‘Idea’ is. While Frost attempts to give Mark a ‘training in objectivity’ that will destroy in him any natural moral sense, and while Mark tries desperately to find a way out of the moral void into which he is being drawn, he discovers what it means to under-stand.

‘He had never before known what an Idea meant: he had always thought till now that they were things inside one’s own head. But now, when his head was continually attacked and often completely filled with the clinging corruption of the training, this Idea towered up above him-something which obviously existed quite independently of himself and had hard rock surfaces which would not give, surfaces he could cling to.’

This too, I fear, is seldom communicated in the classroom, where opinion reigns supreme. But it has important implications for the way we understand argument.”

— “On Bringing One’s Life to a Point,” by Gilbert Meilaender, First Things, November 1994

The old philosophical conflict between realism and nominalism can, it seems, have life-and-death consequences.  I prefer Plato’s realism, with its “ideas,” such as the idea of seven-ness.  A reductio ad absurdum of nominalism may be found in the Stanford Encyclopedia of Philosophy under Realism:

“A certain kind of nominalist rejects the existence claim which the platonic realist makes: there are no abstract objects, so sentences such as ‘7 is prime’ are false….”

The claim that 7 is not prime is, regardless of its motives, dangerously stupid.

The New York Lottery evening number
for All Souls’ Day, Nov. 2, 2005, was

007.

Related material:

Entries for Nov. 1, 2005 and
the song Planned Obsolescence
by the 10,000 Maniacs

(Hope Chest:
The Fredonia Recordings)

Saturday, August 20, 2005

Saturday August 20, 2005

Filed under: General — m759 @ 2:07 PM
Truth vs. Bullshit

Background:
For an essay on the above topic
from this week’s New Yorker,
click on the box below.

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Representing truth:

The image “http://www.log24.com/log/pix05B/050820-Goldstein.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein

Representing bullshit:

The image “http://www.log24.com/log/pix05B/050820-Doxiadis.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis

Goldstein’s truth:

Gödel was a Platonist who believed in objective truth.

See Rothstein’s review of Goldstein’s new book Incompleteness.

Doxiadis’s bullshit:

Gödel, along with Darwin, Marx, Nietzsche, Freud, Einstein, and Heisenberg, destroyed a tradition of certainty that began with Plato and Euclid.

“Examples are the stained-glass
windows of knowledge.” — Nabokov

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: General,Geometry — m759 @ 8:16 AM

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

Wednesday, August 3, 2005

Wednesday August 3, 2005

Filed under: General — m759 @ 2:02 PM

Epiphany Term

“In Epiphany Term, 1942, C.S. Lewis delivered the Riddell Memorial Lectures… in….  the University of Durham….  He delivered three lectures entitled ‘Men without Chests,’ ‘The Way,’ and ‘The Abolition of Man.’  In them he set out to attack and confute what he saw as the errors of his age. He started by quoting some fashionable lunacy from an educationalists’ textbook, from which he developed a general attack on moral subjectivism.  In his second lecture he argued against various contemporary isms, which purported to replace traditional objective morality.  His final lecture, ‘The Abolition of Man,’ which also provided the title of the book published the following year, was a sustained attack on hard-line scientific anti-humanism. The intervening fifty years have largely vindicated Lewis.”

— J. R. Lucas, The Restoration of Man

See also Log24,
Epiphany 2003.
 

Friday, July 29, 2005

Friday July 29, 2005

Filed under: General — Tags: — m759 @ 4:44 AM
Anatomy of a Death

From today's New York Times:

The image “http://www.log24.com/log/pix05A/050729-Held.jpg” cannot be displayed, because it contains errors.

From the Washington Post
:

"Al Held, an American artist who painted large-scale abstract works… was found dead July 27, floating in a swimming pool at his villa…. The cause of death was not reported, but Italian police said he died of natural causes. He was 76."

From the Associated Press
,
filed at 4:34 PM ET July 27, 2005:

"Held once described his work this way: 'Historically, the priests and wise men believed that it was the artist's job to make images of heaven and hell believable, even though nobody had experienced these places.'

'Today,' he went on, 'scientists talk about vast worlds and universes that the senses cannot experience. The purpose of the nonobjective artist is to create these images.'"

Another view:

"Most modern men do not believe in hell because they have not been there."
— Review of Malcolm Lowry's novel Under the Volcano (1947)

Related material:

The Four Last Things.
 

  Hollywood images:
 

The image “http://www.log24.com/log/pix05A/050729-Bass5.jpg” cannot be displayed, because it contains errors.

And from Mathematics and Narrative:

By Their Fruits

Today's (July 22) birthdays:
Don Henley and Willem Dafoe

The image “http://www.log24.com/log/pix05A/050722-Fruits.jpg” cannot be displayed, because it contains errors.

Related material:

Mathematics and Narrative
,

Crankbuster.

"And the fruit is rotten;
the serpent's eyes shine
as he wraps around the vine
in the Garden of Allah."
 

Saturday, June 4, 2005

Saturday June 4, 2005

Filed under: General,Geometry — m759 @ 7:00 PM
  Drama of the Diagonal
  
   The 4×4 Square:
  French Perspectives

Earendil_Silmarils:
The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.
  
   Les Anamorphoses:
 
   The image “http://www.log24.com/log/pix05A/050604-DesertSquare.jpg” cannot be displayed, because it contains errors.
 
  “Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d’horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4).”
   _______________________________
  
  Serge Mehl,
   Perspective &
  Géométrie Projective:
  
   “… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s’opposait à la géométrie
   euclidienne: élémentaire
  
  La géométrie projective, certes supérieure
   car assez ardue, permet d’établir
   de façon élégante des résultats de
   la géométrie élémentaire.”
  
  Similarly…
  
  Finite projective geometry
  (in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
  quilt-pattern symmetry
  and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.
  
  Other Related Material…
  
   from algebra rather than
   geometry, and from a German
   rather than from the French:  

This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

 Weyl also says that the profound branch
of mathematics known as Galois theory

   “… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
    finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory.”
  — Weyl, Symmetry,
   Princeton U. Press, 1952
  
   Metaphor and Algebra…  

“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.” 

   — attributed, in varying forms, to
   Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see  

  “Symmetry invariance
  in a diamond ring,”

  A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

  
More on Max Black…

“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”

— Paul Thompson, University College, Oxford,
    The Nature and Role of Intuition
     in Mathematical Epistemology

  A New Slant…  

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known.  The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.”  The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

Saturday, May 14, 2005

Saturday May 14, 2005

Filed under: General — m759 @ 1:00 PM
Powers,
continued

Today’s New York Times:

“Horton Marlais Davies, Putnam professor emeritus of religion at Princeton and an author of many books about church history, died on Wednesday at his home in Princeton, N.J. He was 89…. Dr. Davies specialized in the impact of Christianity on the arts.”

The image “http://www.log24.com/log/pix05/050514-Cover2.jpg” cannot be displayed, because it contains errors.

A book edited by Horton Davies,
apparently first published by Eerdmans
at Grand Rapids, Michigan, in 1990

The Catholic Encyclopedia (1908) on the communion of saints:

“One cannot read the parables of the kingdom (Matt., xiii) without perceiving its corporate nature and the continuity which links together the kingdom in our midst and the kingdom to come. The nature of that communion, called by St. John a fellowship with one another (‘a fellowship with us’ — I John, i, 3) because it is a fellowship with the Father, and with his Son, and compared by him to the organic and vital union of the vine and its branches (John, xv), stands out….”
 
Related material:

Religious art in the entry Art History of 11 AM Wednesday, May 11, the date of Davies’s death.  See also the following direct and indirect links from that entry:

To a cruciform artifact from the current film Kingdom of Heaven, to an entry quoting John xv, Nine is a Vine, and to Art Theory for Yom Kippur.

For less-religious material on the number nine, see the entries and links in the Log24 archive for June 17-30, 2004.

From Rosalind Krauss, “Grids”:

“If we open any tract–
Plastic Art and Pure Plastic Art
or The Non-Objective World,
for instance– we will find that
Mondrian and Malevich are not
discussing canvas or pigment or
graphite or any other form of
matter.  They are talking about
Being or Mind or Spirit.”

Amen.

Friday, September 17, 2004

Friday September 17, 2004

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

God is in…
The Details

From an entry for Aug. 19, 2003 on
conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

On Harvard and psychiatry: see

The Crimson Passion:
A Drama at Mardi Gras

(February 24, 2004)

This is a reductio ad absurdum of the Harvard philosophy so eloquently described by Alston Chase in his study of Harvard and the making of the Unabomber, Ted Kaczynski.  Kaczynski's time at Harvard overlapped slightly with mine, so I may have seen him in Cambridge at some point.  Chase writes that at Harvard, the Unabomber "absorbed the message of positivism, which demanded value-neutral reasoning and preached that (as Kaczynski would later express it in his journal) 'there is no logical justification for morality.'" I was less impressed by Harvard positivism, although I did benefit from a course in symbolic logic from Quine.  At that time– the early 60's– little remained at Harvard of what Robert Stone has called "our secret culture," that of the founding Puritans– exemplified by Cotton and Increase Mather.

From Robert Stone, A Flag for Sunrise:

"Our secret culture is as frivolous as a willow on a tombstone.  It's a wonderful thing– or it was.  It was strong and dreadful, it was majestic and ruthless.  It was a stranger to pity.  And it's not for sale, ladies and gentlemen."

Some traces of that culture:

A web page
in Australia:

A contemporary
Boston author:

Click on pictures for details.

A more appealing view of faith was offered by PBS on Wednesday night, the beginning of this year's High Holy Days:

Armand Nicholi: But how can you believe something that you don't think is true, I mean, certainly, an intelligent person can't embrace something that they don't think is true — that there's something about us that would object to that.

Jeremy Fraiberg: Well, the answer is, they probably do believe it's true.

Armand Nicholi: But how do they get there? See, that's why both Freud and Lewis was very interested in that one basic question. Is there an intelligence beyond the universe? And how do we answer that question? And how do we arrive at the answer of that question?

Michael Shermer: Well, in a way this is an empirical question, right? Either there is or there isn't.

Armand Nicholi: Exactly.

Michael Shermer: And either we can figure it out or we can't, and therefore, you just take the leap of faith or you don't.

Armand Nicholi: Yeah, now how can we figure it out?

Winifred Gallagher: I think something that was perhaps not as common in their day as is common now — this idea that we're acting as if belief and unbelief were two really radically black and white different things, and I think for most people, there's a very — it's a very fuzzy line, so that —

Margaret Klenck: It's always a struggle.

Winifred Gallagher: Rather than — I think there's some days I believe, and some days I don't believe so much, or maybe some days I don't believe at all.

Doug Holladay: Some hours.

Winifred Gallagher: It's a, it's a process. And I think for me the big developmental step in my spiritual life was that — in some way that I can't understand or explain that God is right here right now all the time, everywhere.

Armand Nicholi: How do you experience that?

Winifred Gallagher: I experience it through a glass darkly, I experience it in little bursts. I think my understanding of it is that it's, it's always true, and sometimes I can see it and sometimes I can't. Or sometimes I remember that it's true, and then everything is in Technicolor. And then most of the time it's not, and I have to go on faith until the next time I can perhaps see it again. I think of a divine reality, an ultimate reality, uh, would be my definition of God.

Winifred
Gallagher

Sangaku

Gallagher seemed to be the only participant in the PBS discussion that came close to the Montessori ideals of conciseness, simplicity, and objectivity.  Dr. Montessori intended these as ideals for teachers, but they seem also to be excellent religious values.  Just as the willow-tombstone seems suited to Geoffrey Hill's style, the Pythagorean sangaku pictured above seems appropriate to the admirable Gallagher.

Tuesday, April 6, 2004

Tuesday April 6, 2004

Filed under: General — Tags: , — m759 @ 10:00 PM

Ideas and Art, Part III

The first idea was not our own.  Adam
In Eden was the father of Descartes…

— Wallace Stevens, from
Notes Toward a Supreme Fiction

“Quaedam ex his tanquam rerum imagines sunt, quibus solis proprie convenit ideae nomen: ut cùm hominem, vel Chimaeram, vel Coelum, vel Angelum, vel Deum cogito.”

Descartes, Meditationes III, 5

“Of my thoughts some are, as it were, images of things, and to these alone properly belongs the name idea; as when I think [represent to my mind] a man, a chimera, the sky, an angel or God.”

Descartes, Meditations III, 5

Begin, ephebe, by perceiving the idea
Of this invention, this invented world,
The inconceivable idea of the sun.

You must become an ignorant man again
And see the sun again with an ignorant eye
And see it clearly in the idea of it.

— Wallace Stevens, from
Notes Toward a Supreme Fiction

“… Quinimo in multis saepe magnum discrimen videor deprehendisse: ut, exempli causâ, duas diversas solis ideas apud me invenio, unam tanquam a sensibus haustam, & quae maxime inter illas quas adventitias existimo est recensenda, per quam mihi valde parvus apparet, aliam verò ex rationibus Astronomiae desumptam, hoc est ex notionibus quibusdam mihi innatis elicitam, vel quocumque alio modo a me factam, per quam aliquoties major quàm terra exhibetur; utraque profecto similis eidem soli extra me existenti esse non potest, & ratio persuadet illam ei maxime esse dissimilem, quae quàm proxime ab ipso videtur emanasse.”

Descartes, Meditationes III, 11

“… I have observed, in a number of instances, that there was a great difference between the object and its idea. Thus, for example, I find in my mind two wholly diverse ideas of the sun; the one, by which it appears to me extremely small draws its origin from the senses, and should be placed in the class of adventitious ideas; the other, by which it seems to be many times larger than the whole earth, is taken up on astronomical grounds, that is, elicited from certain notions born with me, or is framed by myself in some other manner. These two ideas cannot certainly both resemble the same sun; and reason teaches me that the one which seems to have immediately emanated from it is the most unlike.”

Descartes, Meditations III, 11

“Et quamvis forte una idea ex aliâ nasci possit, non tamen hîc datur progressus in infinitum, sed tandem ad aliquam primam debet deveniri, cujus causa sit in star archetypi, in quo omnis realitas formaliter contineatur, quae est in ideâ tantùm objective.”

Descartes, Meditationes III, 15

“And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality [or perfection] that is found objectively [or by representation] in these ideas is contained formally [and in act].”

Descartes, Meditations III, 15

Michael Bryson in an essay on Stevens’s “Notes Toward a Supreme Fiction,”

The Quest for the Fiction of the Absolute:

“Canto nine considers the movement of the poem between the particular and the general, the immanent and the transcendent: “The poem goes from the poet’s gibberish to / The gibberish of the vulgate and back again. / Does it move to and fro or is it of both / At once?” The poet, the creator-figure, the shadowy god-figure, is elided, evading us, “as in a senseless element.”  The poet seeks to find the transcendent in the immanent, the general in the particular, trying “by a peculiar speech to speak / The peculiar potency of the general.” In playing on the senses of “peculiar” as particular and strange or uncanny, these lines play on the mystical relation of one and many, of concrete and abstract.”

Brian Cronin in Foundations of Philosophy:

“The insight is constituted precisely by ‘seeing’ the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight.”

— From Ch. 2, Identifying Direct Insights

Michael Bryson in an essay on Stevens’s “Notes Toward a Supreme Fiction“:

“The fourth canto returns to the theme of opposites. ‘Two things of opposite natures seem to depend / On one another . . . . / This is the origin of change.’  Change resulting from a meeting of opposities is at the root of Taoism: ‘Tao produced the One. / The One produced the two. / The two produced the three. / And the three produced the ten thousand things’ (Tao Te Ching 42) ….”

From an entry of March 7, 2004

From the web page

Introduction to the I Ching–
By Richard Wilhelm
:

“He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the ‘great primal beginning’ of all that exists, t’ai chi — in its original meaning, the ‘ridgepole.’ Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t’ai chi was represented by the circle divided into the light and the dark, yang and yin,

.

This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites.”

The t’ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that

“W.J.T. Mitchell calls ‘iconology’
a study of the ‘logos’
(the words, ideas, discourse, or ‘science’)
of ‘icons’ (images, pictures, or likenesses).
It is thus a ‘rhetoric of images’
(Iconology: Image, Text, Ideology, p. 1).”

A variation on the t’ai chi symbol appears in a log24.net entry for March 5:

The Line,
by S. H. Cullinane

See too my web page Logos and Logic, which has the following:

“The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi, was the first splendor in mathematics.”

— Simone Weil, Oeuvres Choisies, ed. Quarto, Gallimard, 1999, p. 100

 Logos Alogos,
by S. H. Cullinane 

In the conclusion of Section 3, Canto X, of “Notes,” Stevens says

“They will get it straight one day
at the Sorbonne.
We shall return at twilight
from the lecture
Pleased that
the irrational is rational….”

This is the logoi alogoi of Simone Weil.

In “Notes toward a Supreme Fiction,”
Wallace Stevens lists three criteria
for a work of the imagination:

It Must Be Abstract

The Line,
by S.H. Cullinane 

It Must Change

 The 24,
by S. H. Cullinane

It Must Give Pleasure

Puzzle,
by S. H. Cullinane

Related material:

Logos and Logic.

 

Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: — m759 @ 3:24 PM

Finite Relativity

Today is the 18th birthday of my note

The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Tuesday, February 3, 2004

Tuesday February 3, 2004

Filed under: General — m759 @ 11:11 AM

The Quality with No Name

And what is good, Phædrus,
and what is not good…
Need we ask anyone to tell us these things?

— Epigraph to
   Zen and the Art of Motorcyle Maintenance

Brad Appleton discusses a phrase of Christopher Alexander:

“The ‘Quality Without A Name‘ (abbreviated as the acronym QWAN) is the quality that imparts incommunicable beauty and immeasurable value to a structure….

Alexander proposes the existence of an objective quality of aesthetic beauty that is universally recognizable. He claims there are certain timeless attributes and properties which are considered beautiful and aesthetically pleasing to all people in all cultures (not just ‘in the eye of the beholder’). It is these fundamental properties which combine to generate the QWAN….”

See, too, The Alexander-Pirsig Connection.

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,

Framing
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.

Sunday, October 5, 2003

Sunday October 5, 2003

Filed under: General — Tags: — m759 @ 5:09 AM

At Mount Sinai:
Art Theory for Yom Kippur

From the New York Times of Sunday, October 5, 2003 (the day that Yom Kippur begins at sunset):

"Rabbi Ephraim Oshry, whose interpretations of religious law helped sustain Lithuanian Jews during Nazi occupation…. died on Sept. 28 at Mount Sinai Hospital in Manhattan. He was 89."

For a fictional portrait of Lithuanian Jews during Nazi occupation, see the E. L. Doctorow novel City of God.

For meditations on the spiritual in art, see the Rosalind Krauss essay "Grids."   As a memorial to Rabbi Oshry, here is a grid-based version of the Hebrew letter aleph:


Rabbi Oshry


Aleph

Click on the aleph for details.

"In the garden of Adding,
 Live Even and Odd…."   
— The Midrash Jazz Quartet in
       City of God, by E. L. Doctorow

Here are two meditations
on Even and Odd for Yom Kippur:

Meditation I

From Rosalind Krauss, "Grids":

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example, we could think about Ad Reinhardt who, despite his repeated insistence that 'Art is art,' ended up by painting a series of black nine-square grids in which the motif that inescapably emerges is a Greek cross.  There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."

Meditation II

Here, for reference, is a Greek cross
within a nine-square grid:

 Related religious meditation for
    Doctorow's "Garden of Adding"…

 4 + 5 = 9.

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 PM

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Tuesday, August 19, 2003

Tuesday August 19, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:23 PM

Intelligence Test

From my August 31, 2002, entry quoting Dr. Maria Montessori on conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life

 

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

For the meaning of life, see

The Gospel According to Jill St. John,

whose birthday is today.

"Some sources credit her with an I.Q. of 162."
 

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General,Geometry — m759 @ 11:11 PM

The Transcendent
Signified

“God is both the transcendent signifier
and transcendent signified.”

— Caryn Broitman,
Deconstruction and the Bible

“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”

— Henry P. Mills,
The Significance of Language,
Footnote 2

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato’s (realist) reaction to the sophists (nominalists). What is often called ‘postmodernism’ is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth.”

Simon Blackburn, Think,
Oxford University Press, 1999, page 268

The question of universals is still being debated in Paris.  See my July 25 entry,

A Logocentric Meditation.

That entry discusses an essay on
mathematics and postmodern thought
by Michael Harris,
professor of mathematics
at l’Université Paris 7 – Denis Diderot.

A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions.  Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective.  In other words, pi can be identified algebraically with X, the variable par excellence.”

Harris illustrates this with
an X in a rectangle:

For the complete passage, click here.

If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey.  On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”

Kubrick’s
monolith

Harris’s
slab

For a more serious discussion of deconstruction and Christian theology, see

Walker Percy’s Semiotic.

Friday, July 18, 2003

Friday July 18, 2003

Filed under: General,Geometry — m759 @ 4:09 PM

Hideous Strength

On a Report from London:

Assuming rather prematurely that the body found in Oxfordshire today is that of David Kelly, Ministry of Defence germ-warfare expert and alleged leaker of information to the press, the Financial Times has the following:

“Mr Kelly’s death has stunned all the players involved in this drama, resembling as it does a fictitious political thriller.”

Financial Times, July 18,
   2003, 19:06 London time

I feel it resembles rather a fictitious religious thriller… Namely, That Hideous Strength, by C. S. Lewis.  The use of the word “idea” in my entries’ headlines yesterday was not accidental.  It is related to an occurrence of the word in Understanding: On Death and Truth, a set of journal entries from May 9-12.  The relevant passage on “ideas” is quoted there, within commentary by an Oberlin professor:

“That the truth we understand must be a truth we stand under is brought out nicely in C. S. Lewis’ That Hideous Strength when Mark Studdock gradually learns what an ‘Idea’ is. While Frost attempts to give Mark a ‘training in objectivity’ that will destroy in him any natural moral sense, and while Mark tries desperately to find a way out of the moral void into which he is being drawn, he discovers what it means to under-stand.

‘He had never before known what an Idea meant: he had always thought till now that they were things inside one’s own head. But now, when his head was continually attacked and often completely filled with the clinging corruption of the training, this Idea towered up above him-something which obviously existed quite independently of himself and had hard rock surfaces which would not give, surfaces he could cling to.’

This too, I fear, is seldom communicated in the classroom, where opinion reigns supreme. But it has important implications for the way we understand argument.”

— “On Bringing One’s Life to a Point,” by Gilbert Meilaender, First Things,  November 1994

The old philosophical conflict between realism and nominalism can, it seems, have life-and-death consequences.  I prefer Plato’s realism, with its “ideas,” such as the idea of seven-ness.  A reductio ad absurdum of nominalism may be found in the Stanford Encyclopedia of Philosophy under Realism:

“A certain kind of nominalist rejects the existence claim which the platonic realist makes: there are no abstract objects, so sentences such as ‘7 is prime’ are false….”

The claim that 7 is not prime is, regardless of its motives, dangerously stupid… A quality shared, it seems, by many in power these days.

Saturday, July 5, 2003

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 4:17 AM

Elements

In memory of Walter Gropius, founder of the Bauhaus and head of the Harvard Graduate School of Design.  Gropius died on this date in 1969.  He said that

"The objective of all creative effort in the visual arts is to give form to space. … But what is space, how can it be understood and given a form?"

"Alle bildnerische Arbeit will Raum gestalten. … Was ist Raum, wie können wir ihn erfassen und gestalten?"


Gropius

— "The Theory and Organization
of the Bauhaus
" (1923)

I designed the following logo for my Diamond Theory site early this morning before reading in a calendar that today is the date of Gropius's death.  Hence the above quote.

"And still those voices are calling
from far away…"
— The Eagles
 

Stoicheia:

("Stoicheia," Elements, is the title of
Euclid's treatise on geometry.)

Friday, January 24, 2003

Friday January 24, 2003

Filed under: General — Tags: — m759 @ 4:30 AM

Steps

John Lahr on a current production of "Our Town":

"The play's narrator and general master of artifice is the Stage Manager, who gives the phrase 'deus ex machina' a whole new meaning. He holds the script, he sets the scene, he serves as an interlocutor between the worlds of the living and the dead, calling the characters into life and out of it; he is, it turns out, the Author of Authors, the Big Guy himself. It seems, in every way, apt for Paul Newman to have taken on this role. God should look like Newman: lean, strong-chinned, white-haired, and authoritative in a calm and unassuming way—if only we had all been made in his image!"

The New Yorker, issue of Dec. 16, 2002

On this date in 1971, Bill Wilson, co-founder of Alcoholics Anonymous, died. 


Newman


Wilson

"Each person is like an actor who wants to run the whole show; is forever trying to arrange the lights, the ballet, the scenery and the rest of the players in his own way. If his arrangements would only stay put, if only people would do as he wished, the show would be great. Everybody, including himself, would be pleased. Life would be wonderful….

First of all, we had to quit playing God. It didn't work. Next, we decided that hereafter in this drama of life, God was going to be our Director….

When we sincerely took such a position, all sorts of remarkable things followed….

We were now at Step Three."

Alcoholics Anonymous, also known as "The Big Book," Chapter 5 

Postscript of 5:15 AM, after reading the following in the New York Times obituaries:

"Must be a tough objective," says Willie to Joe as they huddle on the side of a road, weapons ready. "Th' old man says we're gonna have th' honor of liberatin' it."

"The old men know when an old man dies."

— Ogden Nash
 

Tuesday, December 3, 2002

Tuesday December 3, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:45 PM

Symmetry, Invariance, and Objectivity

The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.

On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:

"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."

Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):

"Objectivity means invariance with respect to the group of automorphisms."

It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.

I pointed this out on June 10, 2002.

For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).

Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.

Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.

A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:

"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged."  (p. 69)

This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."

Saturday, August 31, 2002

Saturday August 31, 2002

Filed under: General,Geometry — m759 @ 3:36 AM
Today’s birthday: Dr. Maria Montessori

THE MONTESSORI METHOD: CHAPTER VI

HOW LESSONS SHOULD BE GIVEN

“Let all thy words be counted.”
Dante, Inf., canto X.

CONCISENESS, SIMPLICITY, OBJECTIVITY.

…Dante gives excellent advice to teachers when he says, “Let thy words be counted.” The more carefully we cut away useless words, the more perfect will become the lesson….

Another characteristic quality of the lesson… is its simplicity. It must be stripped of all that is not absolute truth…. The carefully chosen words must be the most simple it is possible to find, and must refer to the truth.

The third quality of the lesson is its objectivity. The lesson must be presented in such a way that the personality of the teacher shall disappear. There shall remain in evidence only the object to which she wishes to call the attention of the child….

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale “block design” subtest.

Mathematicians mean something different by the phrase “block design.”

A University of London site on mathematical design theory includes a link to my diamond theory site, which discusses the mathematics of the sorts of visual designs that Professor Pope is demonstrating. For an introduction to the subject that is, I hope, concise, simple, and objective, see my diamond 16 puzzle.

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