Log24

Friday, January 22, 2021

Lichtung!

Filed under: General — Tags: , — m759 @ 2:26 PM

Cards of Identity  continues.

“So I worked the rest of
the feel of the track around
this spacey type sound.”
George Michael on “Father Figure”

Thursday, January 21, 2021

Man Date

Filed under: General — Tags: , , — m759 @ 5:59 PM

The New York Times  this afternoon reports
the death of an author last year
“somewhere between May 2 and May 15.”

“Down and down I go, round and round I go”
Kevin Spacey in the soundtrack album for
“Midnight in the Garden of Good and Evil”

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?

Friday, November 27, 2015

Once Upon a Matrix

Filed under: General,Geometry — Tags: — m759 @ 10:20 PM

Or:  The Strife of Luminosity and Obscurity

(Continued from "Once Upon a Time," November 25, 2015)

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison


Einstein and Geometry

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

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937, 
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

Wednesday, February 8, 2012

Lichtung!

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

(Continued from Dec. 5, 2002)

IMAGE- NY Times: 'For Romney, Night Goes from Bad to Worse,' by Ashley Parker and Michael D. Shear

From Bad…

Shear

Braucht´s noch Text?

To Verse—

lichtung

manche meinen
lechts und rinks
kann man nicht
velwechsern.
werch ein illtum!

by Ernst Jandl

Thursday, February 2, 2012

Die Lichtung

Filed under: General — Tags: , — m759 @ 11:30 AM

See January 4th, 2012.

(This link resulted from an application of Heidegger's
philosophy of "the opening" and "the shining" (Die Lichtung ).)

See also The Shining of May 29.

Update of 12:19 AM Feb. 3, 2012—
The undated (but cached by Google on January 4th, 2012)
unsigned post from a deleted weblog linked to above as
"an application" is also available in a version that is signed
(but still undated).

Saturday, November 12, 2011

The Boys from Uruguay

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

(Continued from September 7th, 2002)

Happy Birthday, Wallace Shawn!

http://www.log24.com/log/pix11C/111112-SpeedOfThought-Shawn.jpg

Shawn in "The Speed of Thought,"
a 2011 film by Evan Oppenheimer.

Uruguay is featured in that film.

See also Lichtung!.

Sunday, July 11, 2010

Language Lab

Filed under: General — Tags: — m759 @ 11:02 AM

From a search in this journal for "Krell"—

Dialogue from an American adaptation of Shakespeare's Tempest

“… Which makes it a gilt-edged priority that one  of us
 gets into that Krell lab and takes that brain boost.”

– Taken from a video, Forbidden Planet Monster Attack

http://www.log24.com/log/pix10A/100711-LanguageLab.jpg

From yesterday's A Manifold Showing

"Propriation gathers the rift-design of the saying and unfolds it
in such a way that it becomes the well-joined structure of a manifold showing."
(p. 415 of Heidegger's Basic Writings, edited by David Farrell Krell,
HarperCollins paperback, 1993)

German versions found on the Web—

„Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des vielfältigen Zeigens.“ 323

323 Heidegger, Weg zur Sprache, S. 259.

"Das Regende im Zeigen der Sage ist das Eignen. Es erbringt das An- und Abwesen in sein jeweilig Eigenes, aus dem dieses sich an ihm selbst zeigt und nach seiner Art verweilt. Das erbringende Eignen, das die Sage als die Zeige in ihrem Zeigen regt, heiße das Ereignen. Es er-gibt das Freie der Lichtung, in die Anwesendes anwähren, aus der Abwesendes entgehen und im Entzug sein Währen behalten kann. Was das Ereignen durch die Sage ergibt, ist nie Wirkung einer Ursache, nicht die Folge eines Grundes. Das erbringende Eignen, das Ereignen, ist gewährender als jedes Wirken, Machen und Gründen. Das Ereignende ist das Ereignis selbst – und nichts außerdem. Das Ereignis, im Zeigen der Sage erblickt, läßt sich weder als ein Vorkommnis noch als ein Geschehen vorstellen, sondern nur im Zeigen der Sage als das Gewährende erfahren. Es gibt nichts anderes, worauf das Ereignis noch zurückführt, woraus es gar erklärt werden könnte. Das Ereignen ist kein Ergebnis (Resultat) aus anderem, aber die Er-gebnis, deren reichendes Geben erst dergleichen wie ein `Es gibt' gewährt, dessen auch noch `das Sein' bedarf, um als Anwesen in sein Eigenes zu gelangen. Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des Vielfältigen Zeigens. Das Ereignis ist das Unscheinbarste des Unscheinbaren, das Einfachste des Einfachen, das Nächste des Nahen und das Fernste des Fernen, darin wir Sterbliche uns zeitlebens aufhalten." 8

8 M. Heidegger: Unterwegs zur Sprache. S. 258 f.

From Google Translate:

"The event brings together the outline of the legend and unfolds it to the structure of the manifold showing."

Wednesday, May 19, 2010

Mathematics and Gestalt

Filed under: General — Tags: , — m759 @ 2:01 AM

"We acknowledge a theorem's beauty
 when we see how the theorem 'fits'
 in its place, how it sheds light around itself,
 like a Lichtung, a clearing in the woods."

 — Gian-Carlo Rota, Indiscrete Thoughts

Here Rota is referring to a concept of Heidegger.

Some context—

"Gestalt Gestell Geviert: The Way of the Lighting,"
 by David Michael Levin in The Philosopher's Gaze

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM
A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings.  Here is a related commentary on a different incarnation of 441.  (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the nm elements not in S.  Then show that G permutes the m elements S into

n(n -1) … (nk + 1)
____________________

m(m – 1) … (mk + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

Thursday, April 17, 2008

Thursday April 17, 2008

Filed under: General — Tags: — m759 @ 8:28 AM
Top Headlines

(at Google News):

  1. Obama, Clinton…
  2. Suicide bomber…

  3. Pope Benedict XVI…

In other words:

  1. The best lack all conviction
  2. while the worst
    Are full of passionate intensity.
  3. Surely some revelation is at hand….

    William Butler Yeats

Revelation for  
April 16, 2008 —
day of the Pennsylvania
Clinton-Obama debate and
 of the Pope’s birthday —

The Pennsylvania Lottery:

PA Lottery April 16, 2008: Mid-day 413, Evening 441

Make of this revelation
what you will.

My own interpretations:
the Lichtung of 4/13 and
the Dickung of page 441
of Heidegger’s
Basic Writings, where
the terms Lichtung and
Dickung are described.

See also “The Shining of
May 29
” (JFK’s birthday).

“By groping toward the light
we are made to realize
how deep the darkness is
around us.”

— Arthur Koestler,  
The Call Girls:
A Tragi-Comedy

Saturday, October 14, 2006

Saturday October 14, 2006

Filed under: General — Tags: — 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, June 14, 2006

Wednesday June 14, 2006

Filed under: General — Tags: — m759 @ 3:48 AM
Shining Appearance

The image “http://www.log24.com/log/pix06A/060614-Heidegger.gif” cannot be displayed, because it contains errors.

Related material on philosophy:

The death of Hollywood agent
Ingo Preminger, brother of
Otto Preminger, on June 7,

the Log24 entry of June 7,
Figures of Speech, and

Lichtung!

Ingo Preminger was also
the producer of the 1970 film MASH.

Related material on brotherhood
and the Korean War:

He Ain’t Heavy.

Sunday, January 4, 2004

Sunday January 4, 2004

Filed under: General — Tags: — m759 @ 2:17 PM

2:17

“… both a new world
And the old made explicit, understood
In the completion of its partial ecstasy,
The resolution of its partial horror.”

— T. S. Eliot, Four Quartets 

Speaking of horror, today’s noon entry has a link to a page that references Stephen King’s The Shining.

On a 1970’s edition of
Stephen King’s The Shining
:

“The page where Danny actually enters room 217 for the first time (King builds to this moment for a long time, it’s one of the more frightening passages in the book), is precisely on page 217. Scared the crap out of me the first time I read it.”

In honor of St. Thomas Stearns Eliot, whose feast day is today, of St. Emil Artin (see entries for St. Emil’s day, 12/20/03), and of Room 217, a check of last year’s 2/17 entries leads to St. Andrea’s weblog, which today, recalling the “white and geometric” prewar Berlin of the 12/20/03 entries, has Andrea looking, with Euclid, on beauty bare.

See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General — Tags: — m759 @ 4:44 PM

ART WARS:

From The New Yorker, issue of March 17, 2003, Clive James on Aldous Huxley:

The Perennial Philosophy, his 1945 book compounding all the positive thoughts of West and East into a tutti-frutti of moral uplift, was the equivalent, in its day, of It Takes a Village: there was nothing in it to object to, and that, of course, was the objection.”

For a cultural artifact that is less questionably perennial, see Huxley’s story “Young Archimedes.”

Plato, Pythagoras, and
the diamond figure

Plato’s Diamond in the Meno
Plato as a precursor of Gerard Manley Hopkins’s “immortal diamond.” An illustration shows the ur-diamond figure.

Plato’s Diamond Revisited
Ivars Peterson’s Nov. 27, 2000 column “Square of the Hypotenuse” which discusses the diamond figure as used by Pythagoras (perhaps) and Plato. Other references to the use of Plato’s diamond in the proof of the Pythagorean theorem:

Huxley:

“… and he proceeded to prove the theorem of Pythagoras — not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself….
‘You see,’ he said, ‘it seemed to me so beautiful….’
I nodded. ‘Yes, it’s very beautiful,’ I said — ‘it’s very beautiful indeed.'”
— Aldous Huxley, “Young Archimedes,” in Collected Short Stories, Harper, 1957, pp. 246 – 247

Heath:

Sir Thomas L. Heath, in his commentary on Euclid I.47, asks how Pythagoreans discovered the Pythagorean theorem and the irrationality of the diagonal of a unit square. His answer? Plato’s diamond.
(See Heath, Sir Thomas Little (1861-1940),
The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publications, New York, 1956.

Other sites on the alleged
“diamond” proof of Pythagoras

Colorful diagrams at Cut-the-Knot

Illustrated legend of the diamond proof

Babylonian version of the diamond proof

For further details of Huxley’s story, see

The Practice of Mathematics,

Part I, by Robert P. Langlands, from a lecture series at the Institute for Advanced Study, Princeton.

From the New Yorker Contributors page for St. Patrick’s Day, 2003:

Clive James (Books, p. 143) has a new collection, As of This Writing: The Essential Essays, 1968-2002, which will be published in June.”

See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.

Thursday, December 5, 2002

Thursday December 5, 2002

Filed under: General — Tags: — m759 @ 3:00 PM

For Otto Preminger’s birthday:

Lichtung!

Today’s symbol-mongering (see my Sept. 7, 2002, note The Boys from Uruguay) involves two illustrations from the website of the Deutsche Schule Montevideo, in Uruguay.  The first, a follow-up to Wallace Stevens’s remarks on poetry and painting in my note “Sacerdotal Jargon” of earlier today, is a poem, “Lichtung,” by Ernst Jandl, with an illustration by Lucia Spangenberg.

Lichtung

manche meinen
lechts und rinks
kann man nicht
velwechsern.
werch ein illtum!

by Ernst Jandl

Lucia Spangenberg, 2002.

The second, from the same school, illustrates the meaning of “Lichtung” explained in my note The Shining of May 29:  

“We acknowledge a theorem’s beauty when we see how the theorem ‘fits’ in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods.”
— Gian-Carlo Rota, page 132 of Indiscrete Thoughts, Birkhauser Boston, 1997

From the Deutsche Schule Montevideo mathematics page, an illustration of the Pythagorean theorem:

Braucht´s noch Text?

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