Monday, February 13, 2017

“The Echo in Plato’s Cave” Continues.

Filed under: General — m759 @ 1:00 PM

The previous post, "Colorful Tales," on the Nov. 28, 2016, death
of one Angus Fletcher, together with the remarks indexed above,
suggest a review from 

The archives of The New York Times  —

Literature as a Way of Life

By Harold Bloom
357 pp. Yale University Press. $32.50.

Sam Tanenhaus is the editor of the Book Review.

A version of this review appeared in print
on May 22, 2011, on Page BR1 of the 
Sunday Book Review with the headline:
'An Uncommon Reader.'"

"By this time, Bloom had burrowed into a cave,
its lamplit forms and shapes merging into
an occult mythos scarcely intelligible
even to other scholars. 'Bloom had an idea,' 
Christopher Ricks said; 'now the idea has him.' 
Cynthia Ozick, meanwhile, called him an 'idol-maker.' 
In contrast to Cleanth Brooks . . . ."

An illustration from "The Echo in Plato's Cave" linked to
in the previous post


Judy Davis in the Marabar Caves

Cynthia Ozick on Bloom —

See also Dharwadker in the previous post and on the Higgs boson.

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.

Monday, October 12, 2015

Space, Time, Matter (continued)

Filed under: General — m759 @ 5:08 PM

Cool Mystery:


Detective Cruz enters Planck's Constant Café in "The Big Bang."

The above images are from a Log24 post of October 5, 2011.

Related material for fans of recreational math and Manil Suri

A book that Amazon.com says was published on that same date —
October 5, 2011 —

Space, Time and Matter 

by Ashay Dharwadker (Author), Vinay Dharwadker  (Author)

Product Details

  • Paperback: 98 pages
  • Publisher: CreateSpace Independent Publishing
                       Platform (October 5, 2011)
  • Language: English
  • Amazon Best Sellers Rank: #7,605,448 in Books 

See as well

Con Vocation  (Log24, Sept. 2, 2014).

Tuesday, September 2, 2014

Con Vocation

Filed under: General — Tags: , — m759 @ 7:09 AM

Con Vocation

A Case in Point — Discuss:

Tuesday, December 2, 2008

Tuesday December 2, 2008

Filed under: General — m759 @ 11:09 AM

A Penny for My Thoughts?
by Maureen Dowd

“If an online newspaper in Pasadena, Calif., can outsource coverage to India, I wonder how long can it be before some guy in Bangalore is writing my column….”

New York Times teaser for a column of Sunday, November 30, 2008 (St. Andrew’s Day)

DH News Service, Bangalore, Tuesday, Dec. 2, 2008:

“Monday evening had a pleasant surprise in store for sky-watchers as the night sky sported a smiley, in the form of a crescent moon flanked by two bright planets Jupiter and Venus…”

Meanwhile, at National Geographic:

Jupiter, Venus, Moon Make “Frown”

A Midrash for Maureen:

The poet’s eye, in fine frenzy rolling,
Doth glance from heaven to earth,
    from earth to heaven;

And as imagination bodies forth
The forms of things unknown,
    the poet’s pen

Turns them to shapes
    and gives to airy nothing

A local habitation and a name.
Such tricks hath strong imagination,
That if it would but
    apprehend some joy,

It comprehends some
    bringer of that joy….”

Related material on Pasadena:
Happy birthday, R. P. Dilworth.

Related material on India:
The Shining of May 29 (2002) and
A Well-Known Theorem (2005).

“Sometimes a line of mathematical research extending through decades can be thought of as one long conversation in which many mathematicians take part. This is fortunately true at present….”

— Barry Mazur in 2000 as quoted today at the University of St. Andrews

Monday, November 24, 2008

Monday November 24, 2008

Filed under: General — m759 @ 8:16 PM

Wow wow wow. Who the heck is this guy?

Tuesday, August 5, 2008

Tuesday August 5, 2008

Filed under: General — Tags: — m759 @ 2:02 PM
Published Today:

Cover of  'The Last Theorem,' a novel by Arthur C. Clarke and Frederik Pohl

The Last Theorem
, a novel by
Arthur C. Clarke and Frederik Pohl

From the publisher's description:

"The Last Theorem is a story of one man’s mathematical obsession, and a celebration of the human spirit and the scientific method. It is also a gripping intellectual thriller….

In 1637, the French mathematician Pierre de Fermat scrawled a note in the margin of a book about an enigmatic theorem: 'I have discovered a truly marvelous proof of this proposition which this margin is too narrow to contain.' He also neglected to record his proof elsewhere. Thus began a search for the Holy Grail of mathematics– a search that didn’t end until 1994, when Andrew Wiles published a 150-page proof. But the proof was burdensome, overlong, and utilized mathematical techniques undreamed of in Fermat’s time, and so it left many critics unsatisfied– including young Ranjit Subramanian, a Sri Lankan with a special gift for mathematics and a passion for the famous 'Last Theorem.'

When Ranjit writes a three-page proof of the theorem that relies exclusively on knowledge available to Fermat, his achievement is hailed as a work of genius, bringing him fame and fortune…."

For a similar third-world fantasy about another famous theorem, see the oeuvre of Ashay Dharwadker.

Note the amazing conclusion of Dharwadker's saga (thus far)–

Dharwadker devises a proof of the four-color theorem that leads to…

Grand Unification
of the Standard Model
with Quantum Gravity!

For further background, see

Ashay Dharwadker
  and Usenet Postings.

Clarke lived in Sri Lanka (formerly Ceylon) from 1956 until his death last March.

For another connection with Sri Lanka, see

Location, Location, Location
(July 13, 2005) and
Bagombo Snuff Box
(May 7, 2006).

Wednesday, July 30, 2008

Wednesday July 30, 2008

Filed under: General,Geometry — m759 @ 11:48 AM
Theories of Everything

Ashay Dharwadker now has a Theory of Everything.
Like Garrett Lisi’s, it is based on an unusual and highly symmetric mathematical structure. Lisi’s approach is related to the exceptional simple Lie group E8.* Dharwadker uses a structure long associated with the sporadic simple Mathieu group M24.



by Ashay Dharwadker


“We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein’s law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with t’Hooft’s table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.”

* See, for instance, “The Scientific Promise of Perfect Symmetry” in The New York Times of March 20, 2007.

Saturday, July 19, 2008

Saturday July 19, 2008

Filed under: General,Geometry — Tags: — 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

Thursday, June 14, 2007

Thursday June 14, 2007

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
A Flag for Sunset

"Kurt Waldheim, the former United Nations Secretary General and President of Austria whose hidden ties to Nazi organizations and war crimes was [sic] exposed late in his career, died today at his home in Vienna. He was 88." —The New York Times this afternoon

Related material:

From a story by
Leonard Michaels
linked to on
Aaron Sorkin's
birthday, June 9:

"Induction and analogy, in which he was highly gifted, were critical to mathematical intelligence.

It has been said that the unexamined life isn't worth living. Nachman wasn't against examining his life, but then what was a life? ….

… As for 'a life,' it was what you read about in newspaper obituaries. He didn't need one. He would return to California and think only about mathematics."


1.  A quotation from George Polya,
     the author of
     Induction and Analogy
     in Mathematics

2.  A quotation from an anonymous
     Internet user signed
     "George Polya"–
     "Steven Cullinane is a Liar."

3.  L'Affaire Dharwadker continues
     (May 31, 2007)

4.  Geometry for Jews

The image “http://log24.com/theory/images/070614-Whirl.gif” cannot be displayed, because it contains errors.

"One two three four,
who are we for?"


Thursday, May 31, 2007

Thursday May 31, 2007

Filed under: General,Geometry — Tags: — m759 @ 8:06 PM

Blitz by anonymous
New Delhi user

From Wikipedia on 31 May, 2007:

Shown below is a list of 25 alterations to Wikipedia math articles made today by user

All of the alterations involve removal of links placed by user Cullinane (myself).

The 122.163… IP address is from an internet service provider in New Delhi, India.

The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews. (See User talk: Cullinane.)

Related material:

Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163…
at Talk: Four color theorem.

May 31, 2007, alterations by

  1. 17:17 Orthogonality (rm spam)
  2. 17:16 Symmetry group (rm spam)
  3. 17:14 Boolean algebra (rm spam)
  4. 17:12 Permutation (rm spam)
  5. 17:10 Boolean logic (rm spam)
  6. 17:08 Gestalt psychology (rm spam)
  7. 17:05 Tesseract (rm spam)
  8. 17:02 Square (geometry) (rm spam)
  9. 17:00 Fano plane (rm spam)
  10. 16:55 Binary Golay code (rm spam)
  11. 16:53 Finite group (rm spam)
  12. 16:52 Quaternion group (rm spam)
  13. 16:50 Logical connective (rm spam)
  14. 16:48 Mathieu group (rm spam)
  15. 16:45 Tutte–Coxeter graph (rm spam)
  16. 16:42 Steiner system (rm spam)
  17. 16:40 Kaleidoscope (rm spam)
  18. 16:38 Efforts to Create A Glass Bead Game (rm spam)
  19. 16:36 Block design (rm spam)
  20. 16:35 Walsh function (rm spam)
  21. 16:24 Latin square (rm spam)
  22. 16:21 Finite geometry (rm spam)
  23. 16:17 PSL(2,7) (rm spam)
  24. 16:14 Translation plane (rm spam)
  25. 16:13 Block design test (rm spam)

The deletions should please Charles Matthews and fans of Ashay Dharwadker’s work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.

There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.

Cullinane 23:28, 31 May 2007 (UTC)

Monday, May 29, 2006

Monday May 29, 2006

Filed under: General — m759 @ 3:00 PM

For John F. Kennedy’s birthday:

The Call Girls

The image “http://www.log24.com/log/pix06A/060529-CallGirls.jpg” cannot be displayed, because it contains errors.

See The Shining of May 29
from 2002
and the references to
the marriage theorem
in Dharwadker’s Alleged Proof
from 2005.

“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,
Random House, 1973, page 118

For related material on
academic darkness, see
Mathematics and Narrative.

Sunday, July 24, 2005

Sunday July 24, 2005

Filed under: General,Geometry — m759 @ 2:56 AM

L’Affaire Dharwadker:

Non-computer proof of 4 color Theorem,
2000 Oct. 13-Nov. 30,
sci.math, 23 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 14,
sci.math, 8 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 15,
comp.misc, 2 posts

Steven Cullinane is a Liar,
2002 Nov. 1-Nov.16,
geometry.research, 2 posts

Four-colour proof claim,
2003 Aug. 10-Sept.1,
sci.math, 9 posts

Proof of 4 colour theorem No computer!!!,
2003 Aug. 10-Aug. 20,
alt.sci.math.combinatorics, 8 posts

Steven Cullinane is a Crank,
2005 July 5-July 21
sci.math, 70 posts

From a Log24 post a year ago today:

“With a holy host of others
     standing ’round me
Still I’m on the dark side
     of the moon…”

— James Taylor

From a Log24 post on July 20 this year:

“And if the band you’re in
 starts playing different tunes
I’ll see you on
the dark side of the moon.”

— Roger Waters

Wednesday, July 7, 2004

Wednesday July 7, 2004

Filed under: General,Geometry — m759 @ 7:00 PM

Beyond Geometry

(Title of current L. A. art exhibit)

John Baez:

What is the difference between topology and geometry?

Geometry you learn in high school; topology in college. So, topology costs more.

A bit more seriously….

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

— Daniel J. Boorstin,
American historian, educator, writer.
Source: The Washington Post,
“The Six O’Clock Scholar,”
by Carol Krucoff (29 Jan. 1984)

For the illusion of knowledge,
see (for instance)
The Importance of Being Nothingness,
by Craig J. Hogan
(American Scientist, Sept.-Oct. 2001).

A bit more seriously…

“These cases are
neither harmless nor amusing.”
— Craig J. Hogan, op. cit.

For example:

“Thanks to Dr. Matrix
for honouring this website
with the Award for Science Excellence
on May 14, 2002 and selecting it
for prominent display in the categories
of Mathematics and Creative Minds.”

See also my notes
On Dharwadker’s Attempted Proof,
November 28, 2000, and
The God-Shaped Hole,
February 21, 2001.

Wednesday, October 2, 2002

Wednesday October 2, 2002

Filed under: General — m759 @ 9:52 AM

A Crackpot with Power

The following is an greatly abbreviated version of a sci.math group thread on an attempted proof of the four-color theorem.

  • Chip Eastham 2000-10-13 :

    There is a nicely presented approach to proving the Four Color Theorem… at the following… site:


  • “Default” 2000-10-13:

    Where in the proof is the hypothesis of “requiring N colors” (not colorable with N-1 colors) used?

  • Michael Varney 2000-10-14:

    (Following some banter) Go play elsewhere if you buy into 4CT crackpot proofs.

  • “Default” 2000-10-14:

    The proposed 4CT proof is hardly crackpot, and may contain some new ideas (or reformulations of old ones).

  • Michael Varney 2000-10-14:

    That’s what all crackpots say. Join the club.

  • David Eppstein 2000-10-14:

    My first-glance reaction is that it’s an amazing collection of undigested chunks of heavy equipment. It seems more designed to confuse any expert (by making sure it contains something the expert doesn’t understand) than to convince anyone of the truth of the 4CT.

  • “Default” 2000-10-15:

    Skimming the proof I did not see any place where the minimality of the chromatic number N was used, nor any explanation of why a 12-fold covering is introduced (other than it fits the numerology needed to rule out a Steiner system). This makes me skeptical about the proof, but it’s hardly crackpot.

The author of this attempted proof, Ashay Dharwadker, is now an editor of the following Open Directory Project categories:

Science: Math: Combinatorics   and
Science: Math: Combinatorics: Graph Theory.

I agree with “Default,” Eppstein, and Varney.

As “Default” notes, the proof is invalid,  since it does not even use the hypotheses of the theorem.  I pointed this out in November 2000 in a sub-page of a website in the Open Directory combinatorics category,

I also agree with Eppstein that Dharwadker’s writing seems “designed to confuse.” 

Finally, I strongly agree with Varney that Dharwadker is a crackpot.  I reluctantly arrived at this conclusion only last night, after learning that

  1. Dharwadker, who formerly had edited only the graph theory Open Directory category, now is a co-editor of its parent category, combinatorics, and that
  2. My website containing a criticism of Dharwadker’s work has been deleted from the Open Directory combinatics listings. This site, “Diamond Theory,” is only incidentally related to Dharwadker’s attempted proof, and has been in the Open Directory combinatorics listing for about two years.  

Crackpots are annoying, but crackpots with power are both contemptible and infuriating.  I am currently trying to rectify the appalling mistake made by whoever appointed Dharwadker to a position of responsibility.

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