Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
See The Righteous Gemstone and Boole vs. Galois.
Update of 9:48 AM EDT Oct. 28 . . .
Related material —
Markdown version uploaded Oct. 28, 2025, to NotebookLM.
From Log24 posts tagged Boole vs. Galois —
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.
See also “A Four-Color Epic” (April 16, 2020).
"… Évariste was born on October 25, 1811."
— Eric Temple Bell, Men of Mathematics
Related material —
But seriously . . .

"… the dominant discourse limits the range
of discussion in each domain…."
— https://americanaffairsjournal.org/2023/11/
the-stagnant-science-mainstream-economics-in-america/
See as well Boole vs. Galois and …
Related elementary mathematics from Google image searches —
Despite the extremely elementary nature of the above tables,
the difference between the binary addition of Boole and that
of Galois seems not to be widely known.
See "The Hunt for Galois October" and "In Memory of a Mississippi Coach."
Last two days of the conference, May 27 and 28, 2022 —
|
27th Friday
9:00 – 10:00 Andrés Villaveces (Univ. Nacional de Colombia):
10:00 – 11:00 Olivia Caramello (Univ. of Insubria; by Zoom): 1:00 – 11:15 Coffee Break
1:15 – 12:15 Mike Shulman (Univ. of San Diego):
12:15 – 1:15 José Gil-Ferez (Chapman Univ.) 1:15 – 2:30 Lunch
2:30 – 3:30 Oumar Wone (Chapman) :
3:30 – 4:30 Claudio Bartocci (Univ. of Genova):
4:30 – 5:30 Christian Houzel (IUFM de Paris): 28th Saturday
9:00 – 10:00 Silvio Ghilardi (Univ. degli Studi, Milano):
10:00 – 11:00 Matteo Viale (Univ. of Turin; by zoom): 11:00 – 11:15 Coffee Break
11:15 – 12:15 Benjamin Collas (RIMS, Kyoto Univ.):
12:15 – 1:15 Closing: general discussion |
Click to enlarge.
Related reading — George Steiner's Fields of Force , on chess in Iceland, and . . .
The New Yorker , article by Sam Knight dated March 28, 2022 —
They went to Björk’s house. She cooked salmon.
She had seen “The Witch” and introduced Eggers
to Sjón, who had written a novel about seventeenth-
century witchcraft in Iceland. When he got home,
Eggers read Sjón’s books. “I was, like, this guy’s
a fucking magician,” Eggers said. “He sees all time,
in time, out of time.”
The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .
Related material from the Web —
Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —
A related anonymous change to Wikipedia today —
The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.
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.
See also “A Four-Color Epic” (April 16, 2020).
(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.
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.
Michael Atiyah on the late Ron Shaw —

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

See also "Romancing the Omega" —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —

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

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:

"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.
"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,
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.
(Continued from previous episodes)
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 —
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
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.
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)

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!) “
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