"Should we arbitrate life and death
at a round table or a square one?"
Vide the geometry in
a post from last summer.
Illustration of a title by George Mackey
Sunday, February 1, 2026
Analogy for the Scottish Café
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Sunday, November 16, 2025
Understanding Media: Analogy
"Say hello to my little friend."
|
https://subslikescript.com/movie/Hurlyburly-119336 — So what do you want to do?
You want to go to your place, You want to go to a sex motel? They got waterbeds.
They got porn I'm hungry. You want a Jack-in-the-Box? I love Jack-in-the-Box. Is that code for something? What? What? Is what code for what? I don't know. I don't know the goddamn code! |
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Monday, October 6, 2025
Analogy Between Analogies
Consider . . .
A. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 labelings of the Fano plane PG(2, 2)
B. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 planes of the Klein quadric in PG(5, 2)
Are A and B nontrivially analogous? If so, how?
Update of 6:58 PM EDT Oct. 7 . . .
Hint:
Use as labels for PG(2, 2) points the seven nonzero vectors in the
3-space over GF(2), expressed as 001, 010, 011, 100, 101, 110, 111.
Then form three seven-digit vectors by taking the first, second, and third
digit in each 3-digit vector. View these seven-digit vectors as points of
the Klein quadric in PG(5, 2).
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Thursday, September 8, 2022
Analogy in Mathematics: Chevron Variations
André Weil in 1940 on analogy in mathematics —
| . "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones." |
"The pleasure comes from the illusion" . . .
Exercise:
Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).
Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —
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Tuesday, June 25, 2019
Thursday, March 1, 2018
The Movement of Analogy: Hume vs. Paz
Hume, from posts tagged "four-set" in this journal —
"The mind is a kind of theatre, where several perceptions
successively make their appearance; pass, repass, glide away,
and mingle in an infinite variety of postures and situations.
There is properly no simplicity in it at one time, nor identity
in different, whatever natural propension we may have
to imagine that simplicity and identity."
Paz, from a search for Paz + Identity in this journal —
|
"At the point of convergence by Octavio Paz, translated by Helen Lane
|
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Tuesday, October 17, 2017
The Movement of Analogy
The title is a phrase by Octavio Paz from today's post
"Status Symbols."
Other phrases from a link target in Sunday's post
The Strength at the Centre —
… a single world
In which he is and as and is are one.
See also Four Dots in this journal.
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Monday, October 2, 2017
The Nut Analogy
Published as the final chapter, Chapter 13, in
Episodes in the History of Modern Algebra (1800-1950) ,
edited by Jeremy J. Gray and Karen Hunger Parshall,
American Mathematical Society, July 18, 2007, pages 301-326.
See also this journal on the above McLarty date —
May 24, 2003: Mental Health Month, Day 24.
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Wednesday, December 23, 2015
Analogy
"An analogy between mathematics and religion is apposite."
— Harvard Magazine review by Avner Ash of
Mathematics without Apologies
(Princeton University Press, January 18, 2015)
See as well Analogies in this journal.
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Thursday, December 18, 2014
Platonic Analogy
(Five by Five continued)
As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged Galois-Plane Models.
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Monday, September 2, 2013
Analogy
From The New York Times Sunday Book Review of Sept. 1, 2013—
|
THE GAMAL Reviewed by Katharine Weber Ten years ago, when Mark Haddon’s “Curious Incident of the Dog in the Night-Time” turned up on the best-seller list and won a number of literary awards, the novel’s autistic narrator beguiled readers with his unconventional point of view. Today, even as controversy surrounds the revised classification of autism in the latest version of the American Psychiatric Association’s Diagnostic and Statistical Manual of Mental Disorders, the quirky yet remarkably perceptive points of view of autistic narrators have become increasingly familiar in every category of fiction, from young adult to science fiction to popular and literary fiction. Like Haddon’s Christopher Boone, the narrator of Ciaran Collins’s remarkable first novel, “The Gamal,” has been encouraged by a mental health professional to write his story for therapeutic purposes. Charlie McCarthy, 25, is known in the West Cork village of Ballyronan as “the gamal,” short for “gamalog,” a term for a fool or simpleton rarely heard beyond the Gaeltacht regions of Ireland. He is in fact a savant, a sensitive oddball whose cheeky, strange, defiant and witty monologue is as disturbing as it is dazzling. … |
The Gamal features a considerable variety of music. See details at a music weblog.
This, together with the narrator's encouragement "by a mental health professional
to write his story for therapeutic purposes" might interest Baz Luhrmann.
See Luhrmann's recent film "The Great Gatsby," with its portrait of
F. Scott Fitzgerald's narrator, and thus Fitzgerald himself, as a sensitive looney.
The Carraway-Daisy-Gatsby trio has a parallel in The Gamal . (Again, see
the music weblog's description.)
The Times reviewer's concluding remarks on truth, lies, and unreliable autistic
narrators may interest some mathematicians. From an Aug. 29 post—
A different gamalog , a website in Mexico, is not entirely unrelated to
issues of lies and truth—
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Friday, December 19, 2025
Who’s Afraid of the Red Lighthouse?
Excerpts from this journal on the dies natalis weekend of the author's late husband,
a UC Berkeley "environmental design" professor —
Saturday, October 8, 2022
|
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Wednesday, October 15, 2025
Sextet Space
“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
Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:
The above fanciful illustration pictures 6*9=54 colored squares on the six
faces of a 3x3x3 cube.
Compare and contrast the Aitchison labeling, not unlike the one above,
of 6*4=24 unit squares (or, equivalently, 24 pips at the squares' centers)
on a 2x2x2 cube.
Now consider how the 8-square "brick" of R. T. Curtis may be colored with
four colors using the 105 ways to partition its eight squares into four 2-sets.
By analogy, the 24 squares on a cube's surface, as above, afford a cubical
space for applying six colors to the sextet partitions (into six 4-sets) of Curtis's
Miracle Octad Generator (MOG), using Aitchson's cubical model (with, of course,
the parts to be moved being pips or squares rather than cuboctahedron edges).
The 4-coloring of Curtis bricks is useful in picturing the Klein correspondence.
Are there similar uses of cube 6-colorings? Or 4-colorings? (Group actions on
a 6-set are of considerable combinatorial and algebraic interest because of
the exceptional outer automorphism of S6.)
For a colored presentation of sextet space modeled with a rectangle,
as in the Curtis MOG, see . . .
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Wednesday, October 8, 2025
Cube-Brick Columns
This post was suggested by yesterday's update to
the "Analogy Between Analogies" post of October 6.
The reason for the above columns . . .
The action of S8 on the rows of an 8-row 3-column matrix
000
001
010
011
100
101
110
111
is intimately connected, via the 30 labelings of a Fano plane
and via the Klein quadric in PG(5, 2), with the action of a
group of order 322,560 on the 16 squares of a 4×4 array.
See Conwell, 1910 [1] and the Log24 tag 105 partitions.
1. Conwell, George M. “The 3-Space PG(3, 2) and Its Group.”
Annals of Mathematics, vol. 11, no. 2, 1910, pp. 60–76.
JSTOR, https://doi.org/10.2307/1967582.
For those who prefer narratives to mathematics: The Cubes.
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Tuesday, June 24, 2025
Scholium for Devs
Related reading:
“Concept (scholastics’ verbum mentis)– theological analogy
of Son’s procession as Verbum Patris, 111-12″
— Index to Joyce and Aquinas, by William T. Noon, S.J.,
Yale University Press 1957, second printing 1963, page 162.
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Tuesday, April 15, 2025
Space Trip
Related drama — Holland Tale, Odious Evening Colors,
The Blue Monkey Diamond, and . . .
|
"At the point of convergence by Octavio Paz, translated by Helen Lane
|
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Wednesday, November 20, 2024
Chevron Space
I prefer a different chevron space . . .
Thursday, September 8, 2022 —
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Wednesday, May 8, 2024
An Antidote to Quanta Magazine
From Quanta Magazine on Monday, May 6, 2024, in
"A Rosetta Stone for Mathematics," by Kevin Hartnett —
" Then he came to the main point of his letter:
He was building such a bridge. He wrote,
'Just as God defeats the devil: this bridge exists.'
The bridge that Weil proposed
is the study of finite fields…."
This is damned nonsense.
From Log24 on June 23, 2005 —
|
In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that “The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two. And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants…. On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.” |
Quanta Magazine's statement:
"The bridge that Weil proposed
is the study of finite fields…."
Here "the study of finite fields" is a contemptibly distorted
dumbing-down of Weil's phrase
"the theory of the field of algebraic functions
over a finite field of constants."
For that topic, see (for instance) . . .
Update at 5:35 PM ET —A different reaction to the Hartnett article —
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Sunday, November 19, 2023
Six Dimensions
"Sharpie, we have condensed six dimensions into four,
then we either work by analogy into six, or we have to use math
that apparently nobody but Jake and my cousin Ed understands.
Unless you can think of some way to project six dimensions into three–
you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done.
Maybe Escher could have done it."
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Wednesday, August 2, 2023
Quale (Rhymes with Folly, not with Tale)
Analogy . . . When A/B = C/D . . .
Illustration from a Log24 post tagged Quale —
— and related material from the date of the above Quale post …
… as well as, from July 31, a version for the institutionalized —
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Wednesday, June 14, 2023
Cheesecake for Princeton
From the previous post . . .
A color analogy — The orange and black (Princeton colors) in the above
conference schedule suggest a recent screengeek image . . .
Related geek lore —
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From Mysticism to Mathematics…
Continued from October 6, 2022 —
A paper from an August 2017 Melbourne conference
on artificial intelligence —
See as well a Log24 search for Boolean functions.
A check on the date of the above paper's presentation —
From this journal on that date —
Happy 10th birthday to the hashtag.
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Friday, April 28, 2023
The Small Space Model
From the previous post, "The Large Language Model,"
a passage from Wikipedia —
"… sometimes large models undergo a 'discontinuous phase shift'
where the model suddenly acquires substantial abilities not seen
in smaller models. These are known as 'emergent abilities,' and
have been the subject of substantial study." — Wikipedia
Compare and contrast
this with the change undergone by a "small space model,"
that of the finite affine 4-space A with 16 points (a Galois tesseract ),
when it is augmented by an eight-point "octad." The 30 eight-point
hyperplanes of A then have a natural extension within the new
24-point set to 759 eight-point octads, and the 322,560 affine
automorphisms of the space expand to the 244,823,040 Mathieu
automorphisms of the 759-octad set — a (5, 8, 24) Steiner system.
For a visual analogue of the enlarged 24-point space and some remarks
on analogy by Simone Weil's brother, a mathematician, see this journal
on September 8 and 9, 2022.
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Thursday, October 6, 2022
From Mysticism to Mathematics
[Klein, 1983] S. Klein.
"Analogy and Mysticism and the Structure of Culture
(and Comments & Reply)"
Current Anthropology , 24 (2):151–180, 1983.
The citation above is from a 2017 paper —
"Analogy-preserving Functions:
A Way to Extend Boolean Samples,"
by M. Couceiro, N. Hug, H. Prade, G, Richard.
26th International Joint Conference on Artificial Intelligence
(IJCAI 2017), Aug. 2017, Melbourne, Australia. pp.1-7, ff.
That 2017 paper discusses Boolean functions .
Some more-recent remarks on these functions
as pure mathematics —
"On the Number of Affine Equivalence Classes
of Boolean Functions," by Xiang-dong Hou,
arXiv:2007.12308v2 [math.CO]. Rev. Aug. 18, 2021.
See also other posts now tagged Analogy and Mysticism.
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Friday, September 9, 2022
The Station
On the Way to Burning Man 2022 —
Chevrons serve as a contrast to the more highly symmetric figures
at left in yesterday morning's Analogy in Mathematics illustration —
Poetry enthusiasts might view the brick at left as
symbolizing the scepter'd isle off the west coast
of Europe, and the gap between as the English
Channel. Mind the gap.
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Wednesday, June 1, 2022
The Yearning
"… every sufficiently good analogy is yearning to become a functor."
— John Baez, 2004: "Lessons from Topological Quantum Field Theory"
Related images — Quoting scripture at The Master's University
on Jan. 31, 2017 —
… and some images (re)posted here on that same date — Jan. 31, 2017:
A related analogy — Between (1) and (2) below —
-
A right triangle A as the sum of two smaller,
inner, right triangles B and C, each similar to A . -
The square on the longest side of A as the sum
of the squares on the longest sides of B and C.
Exercise for Baez fans: Express the above analogy as a functor.
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Monday, April 4, 2022
Pythagoras via Tao, Polya, and Euclid
From a post of April 1 —
A related post by Terry Tao on September 14, 2007 —
The comments on Tao's post contain a reference to Polya's classic
Induction and Analogy in Mathematics . (See pp. 15-17.) Polya notes
on page 15 —
"Generalization, Specialization, and Analogy often concur
in solving mathematical problems. Let us take as an example
the proof of the best known theorem of elementary geometry,
the theorem of Pythagoras. The proof that we shall discuss is
not new; it is due to Euclid himself (Euclid VI, 31)."
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Friday, March 18, 2022
Architectural Review
|
"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind. … Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty. In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics" |
|
Gottschalk Review —
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, The ending of the review — The most striking virtue of the book is its organization. The authors' effort to arrange the exposition in an efficient order, and to group the results together around a few central topics, was completely successful; they deserve to be congratulated on a spectacular piece of workmanship. The results are stated at the level of greatest available generality, and the proofs are short and neat; there is no unnecessary verbiage. The authors have, also, a real flair for the "right" generalization; their definitions of periodicity and almost periodicity, for instance, are very elegant and even shed some light on the classical concepts of the same name. The same is true of their definition of a syndetic set, which specializes, in case the group is the real line, to Bohr's concept of a relatively dense set. The chief fault of the book is its style. The presentation is in the brutal Landau manner, definition, theorem, proof, and remark following each other in relentless succession. The omission of unnecessary verbiage is carried to the extent that no motivation is given for the concepts and the theorems, and there is a paucity of illuminating examples. The striving for generality (which, for instance, has caused the authors to treat uniform spaces instead of metric spaces whenever possible) does not make for easy reading. The same is true of the striving for brevity; the shortest proof of a theorem is not always the most perspicuous one. There are too many definitions, especially in the first third of the book; the reader must at all times keep at his finger tips a disconcerting array of technical terminology. The learning of this terminology is made harder by the authors' frequent use of multiple statements, such as: "The term {asymptotic } {doubly asymptotic } means negatively {or} {and} positively asymptotic." Conclusion: the book is a mine of information, but you sure have to dig for it. — PAUL R. HALMOS |
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Saturday, November 20, 2021
A Saturday Evening Post
|
" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "
— A footnote from page 229 of Sydney Padua's |
Some context: A search in this journal for Lovelace.
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Tuesday, September 14, 2021
Bullshit Studies: Plato vs. Peirce, or Cornerstone vs. Keystone
From a post of August 13, "The Divided Square" —
From a post of April 25, 2016, "Peirce's Accounts of the Universe" —
For the Peirce arrangement, see the imaginary film
reviewed in last night's post Packend! .
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