Log24

The previous post's Lewis Carroll cover,
modified to illustrate Plato's diamond —

See also "To Forge
a Head" (Oct. 27).

A supplement to yesterday's post on variation of an eidos—

Enlarge.

For example, Plato's diamond as an object to be transformed —

Versions of the transformed object —

See also The 4×4 Relativity Problem in this journal.

See also the previous post.

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia —

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

… Before Derrida's writings on Plato and on inscription

A remark by the late William Harris:

"Scholarship has many dark ages, and they do not all fall
in the safe confines of remote antiquity."

For more about Harris, see the previous post.

Discussing an approach to solving a geometrical problem 
from section 86e of the Meno , Harris wrote that

"… this is a very important element of method and purpose,
one which must be taken with great seriousness and respect.
In fact it is as good an example of the master describing for us
his method as Plato ever gives us. Tricked by the appearance
of brevity and unwilling to follow through Plato's thought on
the road to Euclid, we have garbled or passed over a unique
piece of philosophical information."

Harris, though not a geometer, was an admirable man.
His remark on the Meno  method is itself worthy of respect.

In memory of Harris, Plato, and pre-Derrida scholarship, here
are some pages from 1961 on the problem Harris discussed.

A pair of figures from the 1961 pages indicates how one view of the
section 86e problem (at right below) resembles the better-known 
demonstration earlier in the Meno  of how to construct
a square of area 2 —

… Or:  A Funny Thing Happened
     on the Way to the Embedding

This journal on the morning of Saturday, Dec. 8, 2012:

Marilyn Monroe and her music coach in 1954,
from last night's online New York Times :

" 'We were very close to making love; I don’t remember
the stage we were at, but I would say half-dressed,'
Mr. Schaefer recalled. He added: 'And all of a sudden
for some reason, Marilyn got these vibrations, and
we went over to the window….' "  more »

"Mr. Schaefer died on Saturday at 87 at his home in
Fort Lauderdale, Fla. ….

He [had] coached Monroe through 'Diamonds Are
a Girl’s Best Friend,' her signature number in the
1953 movie 'Gentlemen Prefer Blondes' (he arranged
the music as well)…."

Perhaps on Saturday she returned the favor.

(Continued from In Memoriam (Aug. 22), Chapman's Homer (Aug. 23),
and this morning's Colorful Tale)

An informative, but undated, critique of the late Marvin W. Meyer
by April D. DeConick at the website of the Society of Biblical Literature
appeared in more popular form in an earlier New York Times
op-ed piece, "Gospel Truth," dated Dec. 1, 2007.

A check, in accord with Jungian synchronicity, of this  journal
on that date yields a quotation from Plato's Phaedrus  —

"The soul or animate being has the care of the inanimate."

Related verses from T. S. Eliot's Four Quartets —

"The detail of the pattern is movement."

"So we moved, and they, in a formal pattern."

Some background from pure mathematics (what the late
William P. Thurston called "the theory of formal patterns")—

The Animated Diamond Theorem.

Suggested by an Oct. 18 piece in the Book Bench section
of the online New Yorker  magazine—

Related material suggested by the "Shouts and Murmurs" piece
in The New Yorker , issue dated Oct. 24, 2011—

"a series of e-mails from a preschool teacher planning to celebrate
the Day of the Dead instead of Halloween…"

A search for Coxeter + Graveyard in this journal yields…

Here the tombstone says "GEOMETRY… 600 BC — 1900 AD… R.I.P."

A related search for Plato + Tombstone yields an image from July 6, 2007…

Here Plato's poems to Aster suggested
the "Star and Diamond" tombstone.

The eight-rayed star is an ancient symbol of Venus
and the diamond is from Plato's Meno .

The star and diamond are combined in a figure from
12 AM on September 6th, 2011—

From math16.com—

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

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

"Exhibit A is certainly Ljubljana…."

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. (2) The theorems of Euclidean geometry are diamonds. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

Happy Ending

Part I —
Plato's
Tombstone

Part II —
Star and Diamond
United

(See previous post and
a note on design.)

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

See also Plato's Code and
 Plato Thanks the Academy.

Hoax and Hype 
Four Years Ago Today—

There is Plato's diamond—

and there is diamond theory—

… but there is no "Plato's Diamond Theory."

See, however, today's noon entry, "Plato's Code."

"You gotta be true to your code…" —Sinatra

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:

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

“… he triumphed again with The Goldberg Variations. Mr Jay, assisted by Goldberg, a concentration camp survivor, is rehearsing a montage of biblical scenes in Jerusalem. It is inspired satire, laced with Jewish and Christian polemics, sparkling wit and dazzlingly simple effects. For Golgotha a stagehand brings on three crosses. ‘Just two,’ says Jay. ‘The boy is bringing his own.’ Tabori often claimed that the joke was the most perfect literary form.”