Thursday, April 20, 2017

Stone Logic

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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.

Tuesday, August 16, 2016

Midnight Narrative

Filed under: Uncategorized — 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:

Sunday, August 14, 2016

The Boole-Galois Games

Filed under: Uncategorized — 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.

Saturday, July 30, 2016


Filed under: Uncategorized — 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: Uncategorized — 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) —

Thursday, April 14, 2016

One Funeral at a Time

Filed under: Uncategorized — 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: Uncategorized — 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 —

Monday, January 11, 2016

Space Oddity

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — m759 @ 1:01 PM

Bleecker Street logo —

Click image for some background.

Related remarks on mathematics:

Boole vs. Galois

Monday, November 2, 2015

The Devil’s Offer

Filed under: Uncategorized — 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: Uncategorized — 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

"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: Uncategorized — 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: Uncategorized — 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: Uncategorized — m759 @ 7:59 AM

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space —

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

Sunday, September 6, 2015

Elementally, My Dear Watson

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

Galois via Boole

(Courtesy of Intel)

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"
(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: Uncategorized — 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 —

Tuesday, February 5, 2013


Filed under: Uncategorized — Tags: — m759 @ 1:06 PM

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.

Friday, June 23, 2006

Friday June 23, 2006

Filed under: Uncategorized — 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)).


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