Sunday, August 14, 2016

The Boole-Galois Games

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

Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.

Friday, September 11, 2020

Kauffman on Algebra

Filed under: General — Tags: , — m759 @ 11:07 PM

Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry.  See other posts now tagged Boole vs. Galois.

Detail, 8/14/2016 Google image search for 'Galois Boole'

See also “A Four-Color Epic” (April 16, 2020).

Wednesday, November 6, 2019

Soul Snatchers

Filed under: General — m759 @ 11:18 PM

From All Souls' Day 2015

George Boole in image posted on All Souls' Day 2015

Related entertainment —

Invasion of the Soul Snatchers (Wild Palms  review, 1993).

Tuesday, August 13, 2019

Putting the Structure  in Structuralism

Filed under: General — Tags: , — m759 @ 8:34 PM

The Matrix of Lévi-Strauss —

(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée 
, No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)

The structure  of the matrix of Lévi-Strauss —

Illustration from Diamond Theory , by Steven H. Cullinane (1976).

The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.

Thursday, July 19, 2018

The Corrections (Continued)

Filed under: General — m759 @ 8:33 PM

A search for the phrase "nonlinear Boolean algebra" yields few results.

"Nonlinear Boolean functions " seems to be the phrase intended.

On the mathematics of nonlinear Boolean functions —

A memorable death —

For those who prefer narrative to mathematics —

A related image —

Sunday, April 29, 2018


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

From the online New York Times  this afternoon:

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

From this  journal this morning:

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to 
make a fancy amusement-park ride
There had to be more—"

Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"The sensation of folding space . . . ."

Or unfolding:

Click the above unfolded space for some background.

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • “The digital revolution based on the 2 symbols (0,1)”
  • “The algebra of George Boole
  • “Binary codes”
  • “Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Sunday, August 27, 2017

Black Well

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

The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.

Related material —

The Galois Tesseract and, more generally, Binary Coordinate Systems.

Monday, June 12, 2017


Filed under: General,Geometry — m759 @ 1:19 PM

The "bubble" passage in the previous post suggests a review of
a post from December 21, 2006, with the following images —


Update of 11:01 PM ET the same day, June 12, 2017 —

Related material for the Church of Synchronology

From a tech-article series that began on Halloween 2006 and
ended on the date of the above Geometry's Tombstones post —

Compare and contrast (from a post of Feb. 27, 2017) —

“Lord Arglay had a suspicion that the Stone would be
purely logical.  Yes, he thought, but what, in that sense,
were the rules of its pure logic?”

Many Dimensions  (1931), by Charles Williams

See also "The Geometry of Logic:
Finite Geometry and the 16 Boolean Connectives
by Steven H. Cullinane in 2007.

Thursday, April 20, 2017

Stone Logic

Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

See also “Romancing the Omega” —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Related mathematics — Guitart in this journal —

From 'Moving Logic, from Boole to Galois,' by René Guitart, 2005

See also Weyl + Palermo in this journal —


Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM

This post’s title is from the tags of the previous post


The title’s “shift” is in the combined concepts of

Space and Number

From Finite Jest (May 27, 2012):

IMAGE- History of Mathematics in a Nutshell

The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —

Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange

io9 , July 29, 2016

” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego Comic-Con . . . .

‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “

[Typo now corrected, except in a comment.]

Sunday, January 29, 2017

Lottery Hermeneutics

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

For some backstory, see Lottery in this journal,
esp. a post of June 28, 2007:

Real Numbers: An Object Lesson.

One such number, 8775, is suggested by 
a Heinlein short story in a Jan. 25 post.

A search today for that number —

That Jan. 25 post, "For Your Consideration," also mentions logic.

Logic appears as well within a post from the above "8775" date,
August 16, 2016 —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

Related:  Remarks by Charles Altieri on Wittgenstein in
today's previous post.

For remarks by  Wittgenstein related to geometry and logic, see 
(for instance) "Logical space" in "A Wittgenstein Dictionary," by
Hans-Johann Glock (Wiley-Blackwell, 1996).

Friday, November 25, 2016


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

Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:

Hudson in 1905 —

Golomb in 1967 —

It is also likely that some figures illustrating Walsh functions  as
two-color square arrays were published prior to 1976.

Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.

Wednesday, September 21, 2016


Filed under: General — m759 @ 5:01 AM

The phrase "binary opposition" in the previous post suggests
a review of some binary-related concepts —

Boole at Cork

From a post on St. Finbarr's Day 2015

From http://www.chosentwo.com/buffy/quotes/harvest.php

Buffy: So, Giles! Got anything that can make this day any worse?
Giles: How about the end of the world?
Buffy: Knew I could count on you.

Giles: This is what we know. Some sixty years ago, a very old, very powerful vampire came to this shore, not just to feed.
Buffy: He came 'cause this town's a mystical who's it.
Giles: Yes. The Spanish who first settled here called it 'Boca del Infierno'. Roughly translated, 'Hellmouth'. It's a sort of, um, portal between this reality and the next. This vampire hopes to open it.
Buffy: Bring the demons back.
Xander: End of the world.
Willow: But he blew it! Or, I mean, there was an earthquake that swallowed half the town, and him, too.
Giles: You see, opening dimensional portals is a tricky business. Odds are he got himself stuck, rather like a, uh, cork in a bottle.
Xander: And this Harvest thing is to get him out.

Tuesday, August 16, 2016

Midnight Narrative

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

The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —

Boulez and Boole      and

Cross and Boolean lattice.

The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Saturday, July 30, 2016


Filed under: General,Geometry — m759 @ 9:48 AM

"Benedict Cumberbatch Says a Journey
From Fact to Faith Is at the Heart of Doctor Strange

— io9 yesterday

" 'This man comes from a binary universe where it’s all about logic,' 
the actor told us at San Diego Comic-Con . . . .

'And there’s a lot of humor in the collision between Easter [sic ]
mysticism and Western scientific, sort of logical binary.' "

Related material — Strange Awards, April 14, 2016.

I prefer a different sort of journey. See Boole vs. Galois.

Wednesday, July 20, 2016

Weather for Jews

Filed under: General,Geometry — m759 @ 5:00 AM

Somewhere Over the Rainbow

"What I'm aiming for are moments of strong sensation —
moments of total physical experience of the landscape,
when weather just reaches out and sucks you in."

The late Jane Wilson —

See also the previous post and, from the date of Wilson's death,

Geometry for Jews (Continued) —

Sunday, May 8, 2016

The Three Solomons

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

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick’s Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word “conceptual”
in his approach to “conceptual art.”

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

by Sol LeWitt

“The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.”

Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)”

See also the Cullinane models of some small Galois spaces

Some small Galois spaces (the Cullinane models)

Thursday, April 14, 2016

One Funeral at a Time

Filed under: General,Geometry — Tags: — m759 @ 1:37 PM

On this date in 2005, mathematician Saunders Mac Lane died at 95.

Related material —

Max Planck quotations:

Mac Lane on Boolean algebra:

Mac Lane’s summary chart (note the absence of Galois geometry ):

I disagree with Mac Lane’s assertion that “the finite models of
Boolean algebra are dull.”  See Boole vs. Galois in this journal.

Wednesday, January 13, 2016

Geometry for Jews

Filed under: General,Geometry — Tags: — m759 @ 7:45 AM

(Continued from previous episodes)

'Games Played by Boole and Galois'

Boole and Galois also figure in the mathematics of space
i.e. , geometry.  See Boole + Galois in this journal.

Related material, according to Jung’s notion of synchronicity —

Tuesday, January 12, 2016

Lechner’s Beginning

Filed under: General — m759 @ 12:00 AM

Professionally, at least

Click image to enlarge.

See also the previous post, Lechner's End.

For a more up-to-date look at harmonic analysis
and switching functions (i.e., Boolean functions),
see Ryan O'Donnell, Analysis of Boolean Functions ,
Cambridge U. Press, 2014.  Page 40 gives an
informative overview of the history of this field.

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: , — m759 @ 3:15 PM

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.

A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.

“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978

Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981

An illustration from the Lechner paper above —

“There is  such a thing as harmonic analysis of switching functions.”

— Saying adapted from a young-adult novel

Monday, December 28, 2015

ART WARS Continues

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

Combining two headlines from this morning’s
New York Times  and Washington Post , we have

Deceptively Simple Geometries
on a Bold Scale

     Voilà —

Click image for details.

More generally, see
Boole vs. Galois.

Friday, December 25, 2015

Dark Symbol

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

Related material:

The previous post (Bright Symbol) and
a post from Wednesday,
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.

“An analogy between mathematics and religion is apposite.”

— Harvard Magazine  review by Avner Ash of
Mathematics without Apologies
(Princeton University Press, January 18, 2015)

Wednesday, December 23, 2015

Splitting Apart

Filed under: General,Geometry — Tags: — m759 @ 1:01 PM

Bleecker Street logo —

Click image for some background.

Related remarks on mathematics:

Boole vs. Galois

Sunday, December 13, 2015

The Monster as Big as the Ritz

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

“The colorful story of this undertaking begins with a bang.”

— Martin Gardner on the death of Évariste Galois

Monday, November 2, 2015

The Devil’s Offer

Filed under: General,Geometry — Tags: , — m759 @ 11:09 AM

This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space.  From the latter —

” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “

George Boole in image posted on All Souls' Day 2015

Colorful Story

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

"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."

— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575

Or gold letters.

From a search for Seagram in this  journal —

Seagram VO ad, image posted on All Souls's Day 2015

A Seagram 'colorful tale'

"The colorful story of this undertaking begins with a bang."

— Martin Gardner on the death of Évariste Galois

Saturday, October 31, 2015

Raiders of the Lost Crucible

Filed under: General,Geometry — Tags: , — m759 @ 10:15 AM

Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —

Paraconsistent Logic

“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”

This  journal on the date Friday, April 5, 2013 —

The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .

For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching  enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching  studies is,
I maintain, not Boolean algebra  but rather Galois geometry.

See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.

Saturday, October 24, 2015

Two Views of Finite Space

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

The following slides are from lectures on “Advanced Boolean Algebra” —

The small Boolean  spaces above correspond exactly to some small
Galois  spaces. These two names indicate approaches to the spaces
via Boolean algebra  and via Galois geometry .

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

” ‘All you need to do is give me your soul:  give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “

Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977.  That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra  than about Galois geometry .

For later remarks on diamond theory, see finitegeometry.org/sc.

Wednesday, October 21, 2015

Algebra and Space

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space —

"The Quality Without a Name" (Log24, August 26, 2015).

Tuesday, October 20, 2015


Filed under: General — m759 @ 3:33 PM

The New York Times  has a readable, if not informative,
review of a recent controversial account of history —

"For many, it exists in a kind of liminal state,
floating somewhere between fact and mythology."

Jonathan Mahler, online Times  on Oct. 15, 2015

[See Wikipedia on Liminality.]

Mahler begins his review with a statement by the President
on the night of May 1, 2011.

A more easily checked statement quoted here  on that date:

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

A Gestalt  from "Verhexung ," the previous post —

Guitart's statement that the above figure is a "Boolean logical cube"
seems, in the words of the Times , to be "floating somewhere
between fact and mythology."  Discuss.

(My apologies to those who feel that attempting to make sense
of Guitart makes them feel like Vin Diesel in the Dreamworld.)

Monday, October 19, 2015

Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: , — m759 @ 8:48 PM

This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —

“Hexagonal Logic of the Field F8 as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in

The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,

Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.

See also the eightfold cube in this journal.

Sunday, September 6, 2015

Elementally, My Dear Watson

Filed under: General,Geometry — Tags: — m759 @ 9:45 AM

Sarah Larson in the online New Yorker  on Sept. 3, 2015,
discussed Google’s new parent company, “Alphabet”—

“… Alphabet takes our most elementally wonderful
general-use word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm.”

Emma Watson in “The Bling Ring”

This journal, also on September 3 —

Thursday, September 3, 2015

Rings of August

Filed under: Uncategorized — m759 @ 7:20 AM

For the title, see posts from August 2007
tagged Gyges.

Related theological remarks:

Boolean  spaces (old)  vs. Galois  spaces (new)  in 
The Quality Without a Name. . . .

* Actually, Sarah, that would be “phonemes.”

Friday, September 4, 2015

Space Program

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

Galois via Boole

(Courtesy of Intel)

Thursday, September 3, 2015

Rings of August

Filed under: General,Geometry — Tags: , — m759 @ 7:20 AM

For the title, see posts from August 2007 tagged Gyges.

Related theological remarks:

Boolean  spaces (old) vs. Galois  spaces  (new) in
The Quality Without a Name
(a post from August 26, 2015) and the

Related literature:  A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.

Wednesday, August 26, 2015

“The Quality Without a Name”

Filed under: General,Geometry — Tags: , , — m759 @ 8:00 AM

The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander’s book
The Timeless Way of Building  (Oxford University Press, 1979).

A quote from the publisher:

“Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself.”

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of “Boolean space”:

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the “unfolding,” as Christopher Alexander would have it.

These “Boolean” spaces of 1, 2, 4, 8, and 16 points
are also Galois  spaces.  See the diamond theorem —

Sunday, December 14, 2014


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

"In digital circuit theory, combinational logic 
(sometimes also referred to as time-independent logic)
is a type of digital logic which is implemented by
Boolean circuits, where the output is a pure function of
the present input only."

Wikipedia, quoted in this morning's previous post as
commentary on Nabokov's phrase "combinational delight"

"Time past and time future
Allow but a little consciousness.
To be conscious is not to be in time
But only in time can the moment in the rose-garden,
The moment in the arbour where the rain beat,
The moment in the draughty church at smokefall
Be remembered; involved with past and future.
Only through time time is conquered."

— T. S. Eliot in Four Quartets

"I confess I do not believe in time." 

— Vladimir Nabokov in Speak, Memory

Tuesday, February 5, 2013


The previous post discussed some fundamentals of logic.

The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not  the algebra
needed for Galois geometry . See below.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.

See also yesterday’s Universe of Discourse.


Filed under: General — Tags: — m759 @ 12:24 PM

From January 26, 2013

IMAGE- Yahoo CEO Marissa Mayer at Davos and the ontology of entities

Related material: "universe of discourse"

Thursday, September 6, 2012

Decomp Revisited

Filed under: General — m759 @ 1:11 PM


"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time."

— Freeman Dyson (See July 22, 2011)

A Rhetorical Question:

Robert Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer:


Above: Amy Adams in "Sunshine Cleaning"

Related material:

Tuesday, May 1, 2012

What is Truth? (continued)

Filed under: General — m759 @ 11:01 PM

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

Returning to the Walpurgisnacht posts
Decomposition (continued) and
Decomposition– Part III —

Some further background…


(Not  a Scholastic Aptitude Test)

"In computer sciencesatisfiability (often written
in all capitals or abbreviated 
SAT) is the problem
of determining if the variables of a given 
 formula can be assigned in such a way as to
make the formula evaluate to TRUE."

— Wikipedia article Boolean satisfiability problem

For the relationship of logic decomposition to SAT,
see (for instance) these topics in the introduction to—

Advanced Techniques in Logic Synthesis,
Optimizations and Applications* 

Click image for a synopsis.

* Edited by Sunil P. Khatri and Kanupriya Gulati

Friday, September 17, 2010

The Galois Window

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

Yesterday’s excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.

That approach will appeal to few mathematicians, so here is another.

Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace  is a book by Leonard Mlodinow published in 2002.

More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design  (published on September 7, 2010).

A review of Mlodinow’s book on geometry—

“This is a shallow book on deep matters, about which the author knows next to nothing.”
— Robert P. Langlands, Notices of the American Mathematical Society,  May 2002

The Langlands remark is an apt introduction to Mlodinow’s more recent work.

It also applies to Martin Gardner’s comments on Galois in 2007 and, posthumously, in 2010.

For the latter, see a Google search done this morning—


Here, for future reference, is a copy of the current Google cache of this journal’s “paged=4” page.

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron’s web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

See also, in this  journal, Window and Window, continued (July 5 and 6, 2010).

Gardner scoffs at the importance of Galois’s last letter —

“Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers.”
Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society  in March 1899 and February 1909.

Tuesday, June 1, 2010

The Gardner Tribute

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

“It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue.”

Roger Kimball of The New Criterion, May 23, 2010.

The Gardner piece is now online.  It contains…

Gardner’s tribute to Galois—

“Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a ‘personality disorder.’  His anger was
paranoid and unremitting.”

Thursday, May 31, 2007

Thursday May 31, 2007

Filed under: General,Geometry — 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)

Thursday, May 3, 2007

Thursday May 3, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:00 PM
A Web
of Links

"Some postmodern theorists like to talk about the relationship between 'intertextuality' and 'hypertextuality'; intertextuality makes each text a 'mosaic of quotations' [Kristeva, Desire in Language, Columbia U. Pr., 1980, 66] and part of a larger mosaic of texts, just as each hypertext can be a web of links and part of the whole World-Wide Web." —Wikipedia

The image “http://www.log24.com/log/pix07/070503-Tiffany.jpg” cannot be displayed, because it contains errors.

Related material

Day Without Logic,
Introduction to Logic,
The Geometry of Logic,
Structure and Logic,
Spider-Man and Fan:

The image “http://www.log24.com/log/pix07/070503-Devillers.jpg” cannot be displayed, because it contains errors.

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

Wednesday, March 21, 2007

Wednesday March 21, 2007

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

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.

Wednesday, January 3, 2007

Wednesday January 3, 2007

Filed under: General — m759 @ 11:32 AM
The Wanderer:

“What on earth is
a concrete universal?”
Robert M. Pirsig  

Hexagram 56

“James Joyce meant Finnegans Wake to become a universal book. His universe was primarily Dublin, but Joyce believed that the universal can be found in the particular. ‘I always write about Dublin,’ he said to Arthur Power, ‘because if I can get to the heart of Dublin I can get to the heart of all the cities of the world’ (Ellmann 505). He achieved that goal in Ulysses by making Bloom a universal wanderer, the everyman trying to find his way in the labyrinth of the world.” —The Joyce of Science

Related material:

From A Shot at Redemption

The Past as Prologue:
Grand Rapids Revisited

Constantine (cartoon) and Donald Knuth

John Constantine,
cartoon character, and
Donald E. Knuth,
Lutheran mathematician

“…. recent books testify
further to Calvin College’s
unparalleled leadership
in the field of
Christian historiography….”

“I need a photo opportunity,
I want a shot at redemption.
Don’t want to end up a cartoon
In a cartoon graveyard.”

A photo opportunity —

Photo op for Gerald Ford

and a recent cartoon:

Cartoon of Gerald Ford with halo

History, said Stephen….

From Calvin College,
today’s meditation:

Tuesday, June 27, 2006

Tuesday June 27, 2006

Filed under: General,Geometry — m759 @ 10:31 AM
Chinese Jar

In memory of
Irving Kaplansky,
who died on
Sunday, June 25, 2006

“Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

T. S. Eliot

Kaplansky received his doctorate in mathematics at Harvard in 1941 as the first Ph.D. student of Saunders Mac Lane.

From the April 25, 2005, Harvard Crimson:

Ex-Math Prof Mac Lane, 95, Dies

Gade University Professor of Mathematics Barry Mazur, a friend of the late Mac Lane, recalled that [a Mac Lane paper of 1945] had at first been rejected from a lower-caliber mathematical journal because the editor thought that it was “more devoid of content” than any other he had read.

“Saunders wrote back and said, ‘That’s the point,'” Mazur said. “And in some ways that’s the genius of it. It’s the barest, most Beckett-like vocabulary that incorporates the theory and nothing else.”

He likened it to a sparse grammar of nouns and verbs and a limited vocabulary that is presented “in such a deft way that it will help you understand any language you wish to understand and any language will fit into it.”

A sparse grammar of lines from Charles Sanders Peirce (Harvard College, class of 1859):

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

The image “http://www.log24.com/theory/images/PeirceSymbols1.jpg” cannot be displayed, because it contains errors.

It is true of this set of binary connectives, as it is true of logic generally, that (as alleged above of Mac Lane’s category theory) “it will help you understand any language you wish to understand and any language will fit into it.” Of course, a great deal of questionable material has been written about these connectives. (See, for instance, Piaget and De Giacomo.) For remarks on the connectives that are not questionable, see Wittgenstein’s Tractatus Logico-Philosophicus (English version, 1922), section 5.101, and Knuth’s “Boolean Basics” (draft, 2006).

Related entry: Binary Geometry.

Friday, June 23, 2006

Friday June 23, 2006

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

Binary Geometry

There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).


Monday, August 4, 2003

Monday August 4, 2003

Filed under: General — m759 @ 1:00 PM

Venn’s Trinity

Today is the birthday of logician John Venn.

From the St. Andrews History of Mathematics site:

“Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T.” 

Last night’s entry, “A Queer Religion,” gave a Catholic view of the Trinity.  Here are some less interesting but more fruitful thoughts inspired by Venn’s diagram of the Trinity (or, indeed, of any three entities):

“To really know a subject you’ve got to learn a bit of its history….”
John Baez, August 4, 2002

“We both know what memories can bring;
They bring diamonds and rust.”
Joan Baez, April 1975

For the “diamonds” brought by memories of the 28 combinations described above, consider how the symmetric group S8 is related to the symmetries of the finite projective space PG(3,2).  (See Diamond Theory.) 

For the “rust,” consider the following:

“Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt….”
— Matthew 6:19

The letters R, U, S, T in the Venn diagram above are perhaps relevant here, symbolizing, if you will, the earthly confusion of language, as opposed to the heavenly clarity of mathematics.

As for MOTH, see the article Hometown Zeroes (which brings us yet again to the Viper Room, scene of River Phoenix’s death) and the very skillfully designed website MOTHEMATICS.

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