Log24

Tuesday, January 2, 2024

Mathematics Made Absurd:  Domain and Range

Filed under: General — Tags: , — m759 @ 7:54 am

" 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

Monday, May 15, 2023

Boolean Functions Review

Filed under: General — Tags: , — m759 @ 1:25 pm

The previous post included an illustration by Solomon Golomb
from his 1959 paper "On the Classification of Boolean Functions."

This suggests a review of some later work in this area —

This post was suggested by the word "Boolean" in a May 10
ChatGPT response —

In the above, "Boolean algebras" should be "Boolean functions,"
as indicated by Harrison's 1964 remarks.

Death on Beltane

Filed under: General — Tags: , , — m759 @ 2:46 am

"Stencils" from a 1959 paper by Golomb —

Boolean functions illustration by Golomb, 1959

These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).

This  journal on Beltane (May 1), 2016 —

Tuesday, December 13, 2022

In Memory of a Mississippi Coach

Filed under: General — Tags: , — m759 @ 1:10 pm

Remarks in memory of football coach Mike Leach of Mississippi State University

The "Boolean exclusive or" is the same as addition 
in the two-element Galois field GF(2).

Sunday, December 6, 2020

“Binary Coordinates”

Filed under: General — Tags: — m759 @ 3:09 pm

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.

Sunday, November 22, 2020

The Galois-Fano Plane

Filed under: General — Tags: , — m759 @ 9:52 pm

A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —

Related material — The Eightfold Cube.

Update at 10:51 PM ET the same day —

Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .

DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.

The arXiv versions

Sunday, November 15, 2020

Map Methods

Filed under: General — Tags: , — m759 @ 2:04 pm

See also Priority (November 25, 2016).

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).

In Memoriam

Filed under: General — Tags: , , , — m759 @ 2:14 pm
 
From the Vanderbilt University obituary of Vaughan F. R. Jones —

"During the mid-1980s, while Jones was working on a problem in von Neumann algebra theory, which is related to the foundations of quantum mechanics, he discovered an unexpected link between that theory and knot theory, a mathematical field dating back to the 19th century.

Specifically, he found a new mathematical expression—now known as the Jones polynomial—for distinguishing between different types of knots as well as links in three-dimensional space. Jones’ discovery had been missed by topologists during the previous 60 years, and his finding contributed to his selection as a Fields Medalist.

'Now there is an area of mathematics called
quantum topology, which basically followed
from his original work,'

said Dietmar Bisch, professor of mathematics." [Link added.]

Related to Jones's work —

"Topological Quantum Information Theory" at
the website of Louis H. Kauffman —
http://homepages.math.uic.edu/~kauffman/Quanta.pdf.

Kauffman

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.

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.

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 —

http://www.log24.com/log/pix11B/110922-TriquetrumCube.jpg

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.]

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:

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.

Sunday, May 8, 2016

The Three Solomons

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:

 

THE SQUARE AND THE CUBE
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 —

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

"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

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.

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 —

Tuesday, February 5, 2013

Arsenal

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, 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—

http://www.log24.com/log/pix10B/100917-GardnerGalois.jpg

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.”

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)).

Examples:

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