Tuesday, May 5, 2015

Harvard Cinco de Mayo

Filed under: General — Tags: , — m759 @ 1:00 PM

… And Some Not So Live —

"Here was finality indeed, and cleavage!" — Under the Volcano

Sunday, August 30, 2015


Filed under: General,Geometry — Tags: , — m759 @ 11:01 AM

"We tell ourselves stories in order to live." — Joan Didion

A post from St. Augustine's day, 2015, may serve to
illustrate this.

The post started with a look at a painting by Swiss artist
Wolf Barth, "Spielfeld." The painting portrays two
rectangular arrays, of four and of twelve subsquares, 
that sit atop a square array of sixteen subsquares.

To one familiar with Euclid's "bride's chair" proof of the
Pythagorean theorem, "Spielfeld" suggests a right triangle
with squares on its sides of areas 4, 12, and 16.

That image in turn suggests a diagram illustrating the fact
that a triangle suitably inscribed in a half-circle is a right 
triangle… in this case, a right triangle with angles of 30, 60,
and 90 degrees… Thus —

In memory of screenwriter John Gregory Dunne (husband
of Joan Didion and author of, among other things, The Studio
here is a cinematric approach to the above figure.

The half-circle at top suggests the dome of an observatory.
This in turn suggests a scene from the 2014 film "Magic in
the Moonlight."  

As she gazes at the silent universe above
through an opening in the dome, the silent
Emma Stone is perhaps thinking, 
prompted by her work with Spider-Man

"Drop me a line."

As he  gazes at the crack in the dome,
Stone's costar Colin Firth contrasts the vastness 
of the Universe with the smallness of Man, citing 

"the tiny field F2 with two elements."

In conclusion, recall the words of author Norman Mailer
that summarized his Harvard education —

"At times, bullshit can only be countered
with superior bullshit."

Sunday, May 3, 2015

In Memoriam

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

For the author of Dances with Wolves —

'Winter Count,' by Barry Holstun Lopez, cover with shades of gray

Saturday, May 2, 2015

Die Scheinung

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

See also Die Scheinung  in this journal.

Wednesday, April 22, 2015

ART WARS continued

Filed under: General — Tags: — m759 @ 8:29 PM

The previous post mentioned a new mobile, "Triangle Constellation,"
commissioned for the Harvard Art Museums.

Related material (click to enlarge) —

The above review is of an exhibition by the "Constellation" artist,
Carlos Amorales, that opened on Sept. 26, 2008 — "just in time for
Halloween and the Day of the Dead."

See also this  journal on that date.

Purely Aesthetic

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

G. H. Hardy in A Mathematician's Apology —

What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s or Pythagoras’s?

I will not risk more than a few disjointed remarks. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. 

Related material:

Tuesday, October 21, 2014


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

(Night at the Museum continues.)

"Strategies for making or acquiring tools

While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"

— Jamie Cohen-Cole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science  vol. 40, no. 4 (2007): 567-597.

Obituary of a co-founder, in 1960, of the Center for Cognitive Studies at Harvard:

"Disciplinary Exchange" —

In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —

The Kaleidoscope Puzzle,  The Diamond 16 Puzzle
The 2x2x2 Cube, and The 4x4x4 Cube

"Intellectual Economies" —

In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —

This illustrates not geometry, but rather creative capitalism.

Related material from the date of the above Harvard death:  Art Wars.

Saturday, September 20, 2014


Filed under: General — Tags: — m759 @ 9:00 PM

Once again, Harvard defeats Holy Cross.

IMAGE- Joseph Campbell, 'The Inner Reaches of Outer Space,' meditation on the number nine, the Goddess, and the Angelus

See also a related remark by Norman Mailer, and Plan 9 in this journal.

Presumably the Holy Cross defeat will please art theorist Rosalind Krauss (below).

(Click to enlarge.)

Tuesday, December 18, 2012

Monkey Grammar

Filed under: General,Geometry — Tags: , — m759 @ 2:29 AM

For a modern Adam and Eve—

W. Tecumseh Fitch and Gesche Westphal Fitch,
editors of a new four-volume collection titled
Language Evolution  (Feb. 2, 2012, $1,360)—

Related material—

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by
Helen Lane (Kindle edition of
2011-11-07, Kindle locations

The "play of mirrors" link above is my own.

Click on W. Tecumseh Fitch for links to some
examples of mirror-play in graphic design—
from, say, my own work in a version of 1977, not from
the Fitches' related work published online last June—

See also Log24 posts from the publication date
of the Fitches' Language Evolution

Groundhog Day, 2012.

Happy birthday to the late Alfred Bester.

Saturday, December 8, 2012

Defining the Contest…

Filed under: General,Geometry — Tags: , , , , — m759 @ 5:48 AM

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

"Happy Birthday, Noam Chomsky." by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

See Meno Diamond in this journal. For instance, from 
the Feast of Saint Nicholas (Dec. 6th) this year—

The Meno Embedding


For related truths about geometry, see the diamond theorem.

For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).

See esp. the Sept. 11 post,  on a Royal Society paper from July 2012
claiming that

"With the results presented here, we have taken the first steps
in decoding the uniquely human  fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "

The sorts of patterns discussed in the 2012 paper —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Monday, September 17, 2012

Pattern Conception

Filed under: General,Geometry — Tags: , , , — m759 @ 10:00 AM

( Continued from yesterday's post FLT )

Context Part I —

"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….

…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory  (FLT)…."

— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B  (2012) 367, 1925-1932 

Context Part II —

IMAGE- Wikipedia article 'Formal language'

Context Part III —

A four-color theorem describes the mathematics of
general  structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.

Context Part IV —

A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—

"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."

Related material from a literate source—

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

Formal Patterns—

Not formal language theory  but rather
finite projective geometry  provides a graphic grammar
of abstract design

IMAGE- Harvard Crimson ad, Easter Sunday, 2008: 'Finite projective geometry as a graphic grammar of abstract design'

See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.

Sunday, September 16, 2012


Filed under: General,Geometry — Tags: , , — m759 @ 8:28 PM

The "FLT" of the above title is not Fermat's Last Theorem,
but Formal Language Theory (see image below).

In memory of George A. Miller, Harvard cognitive psychologist, who
reportedly died at 92 on July 22, 2012, the first page of a tribute
published  shortly before his death

IMAGE- Design and Formal Language Theory

The complete introduction is available online. It ends by saying—

"In conclusion, the research discussed in this issue
breathes new life into a set of issues that were raised,
but never resolved, by Miller 60 years ago…."

Related material: Symmetry and Hierarchy (a post of 9/11), and
Notes on Groups and Geometry, 1978-1986 .

Happy Rosh Hashanah.

Friday, October 12, 2007

Friday October 12, 2007

Filed under: General — m759 @ 9:10 AM
A Harvard Education
in a Sentence:

“At times, bullshit can
only be countered
with superior bullshit.”

Norman Mailer,
Harvard ’43

Illustration from
today’s Crimson:

Nobel Laureate Morrison
Reads at Opening Event

Friday, October 12, 2007 3:17 AM

From the reserved elegance of Memorial Church to the sweeping grandeur of Sanders Theatre, the Harvard community honored 28th University President Drew G. Faust with two festive events on the eve of her inauguration.

Thursday, October 11, 2007

Thursday October 11, 2007

Filed under: General — m759 @ 9:26 PM
Words and Music
suggested by the recent
Princeton symposium
"Deep Beauty"

The image “http://www.log24.com/log/pix07A/071011-vonNeumann.jpg” cannot be displayed, because it contains errors.

1. From my childhood:

"You remind me of a man."
"What man?"
"The man with the power."
"What power?"
"The power of hoodoo."
"You do."
"Do what?
"Remind me of a man…."

— Dialogue from
"The Bachelor and the
Bobby-Soxer" (1947)

2.  From later years:

"When I was a little boy,
(when I was just a boy)
and the Devil would
call my name
(when I was just a boy)
I'd say 'now who do,
who do you think
you're fooling?'"

Paul Simon, 1973 

"At times, bullshit can
only be countered
with superior bullshit."
— Norman Mailer

(See A Harvard Education
in a Sentence.)

From Plato's Cave:

A description of caveman life
translated from German

John von

"Soon Freud, soon mourning,
Soon Fried, soon fight.
Nevertheless who know this language?"

(Language courtesy of
Google's translation software)

Picture of von Neumann courtesy of
Princeton University Library

More from Rhymin' Simon–

"one funny mofo"–

"Oh, my mama loves,
she loves me,
she get down on her knees
and hug me
like she loves me
like a rock.
She rocks me
like the rock of ages"

Related material:

The previous Log24 entries
of Oct. 7-11, 2007, and
the five Log24 entries
ending with "Toy Soldiers"
(Valentine's Day, 2003).

See also

"Taking Christ to the Movies,"
by Anna Megill, Princeton '06

Saturday, July 21, 2007

Saturday July 21, 2007

Filed under: General — m759 @ 9:45 AM

Death of a Nominalist

“All our words from loose using have lost their edge.” –Ernest Hemingway

(The Hemingway quotation is from the AP’s “Today in History” on July 21, 2007; for the context, see Death in the Afternoon.)

Today seems as good a day as any for noting the death of an author previously discussed in Log24 on January 29, 2007, and January 31, 2007.

Joseph Goguen
died on July 3, 2006. (I learned of his death only after the entries of January 2007 were written. They still hold.)

Goguen’s death may be viewed in the context of the ongoing war between the realism of Plato and the nominalism of the sophists. (See, for instance, Log24 on August 10-15, 2004, and on July 3-5, 2007.)

Joseph A. Goguen, “Ontology, Society, and Ontotheology” (pdf):

“Before introducing algebraic semiotics and structural blending, it is good to be clear about their philosophical orientation. The reason for taking special care with this is that, in Western culture, mathematical formalisms are often given a status beyond what they deserve. For example, Euclid wrote, ‘The laws of nature are but the mathematical thoughts of God.’ Similarly, the ‘situations’ in the situation semantics of Barwise and Perry, which resemble conceptual spaces (but are more sophisticated– perhaps too sophisticated), are considered to be actually existing, real entities [23], even though they may include what are normally considered judgements.5 The classical semiotics of Charles Sanders Peirce [24] also tends towards a Platonist view of signs. The viewpoint of this paper is that all formalisms are constructed in the course of some task, such as scientific study or engineering design, for the heuristic purpose of facilitating consideration of certain issues in that task. Under this view, all theories are situated social entities, mathematical theories no less than others; of course, this does not mean that they are not useful.”

5 The “types” of situation theory are even further removed from concrete reality.

[23] Jon Barwise and John Perry. Situations and Attitudes. MIT (Bradford), 1983.
[24] Charles Sanders Peirce. Collected Papers. Harvard, 1965. In 6 volumes; see especially Volume 2: Elements of Logic.

From Log24 on the date of Goguen’s death:

Requiem for a clown:

“At times, bullshit can only be
countered with superior bullshit.”

Norman Mailer

This same Mailer aphorism was quoted, along with an excerpt from the Goguen passage above, in Log24 this year on the date of Norman Mailer’s birth.  Also quoted on that date:

Sophia. Then these thoughts of Nature are also thoughts of God.

Alfred. Undoubtedly so, but however valuable the expression may be, I would rather that we should not make use of it till we are convinced that our investigation leads to a view of Nature, which is also the contemplation of God. We shall then feel justified by a different and more perfect knowledge to call the thoughts of Nature those of God….

Whether the above excerpt– from Hans Christian Oersted‘s The Soul in Nature (1852)– is superior to the similar remark of Goguen, the reader may decide.

Friday, November 5, 2004

Friday November 5, 2004

Filed under: General — m759 @ 12:00 PM

A Harvard Education
in a Sentence

Harvard alumnus Norman Mailer:

At times, bullshit can only be countered with superior bullshit.

For Harvard bullshit, see
The Crimson Passion.

For superior bullshit, see
Shrine of the Holy Whapping.

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