Recent logic-related posts suggest . . .
"Can you make it any more complicated?"
Sunday, October 15, 2023
“Simplify, simplify.” — Thoreau
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Monday, April 18, 2022
Iconic Simplicity
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Sunday, April 17, 2022
Annals of Iconic Simplicity
The New York Times today has an obituary for
Kevin Lippert, the founder and publisher of
Princeton Architectural Press, who reportedly
died at 63 on March 29, 2022.
“'There was a space between the academic,
theory-heavy M.I.T. Press and the coffeetableism
of Rizzoli,' Mr. Lamster wrote, adding that
Princeton Architectural Press would fill the gap
with 'the voice of the young practitioner.'
Mr. Lippert championed emerging architects.
He published Steven Holl’s seminal architectural
manifesto, 'Anchoring,' in 1989, and wrote the
introduction to the book of the same name.
Mr. Holl, in a tribute to Mr. Lippert on his website,
called him 'a committed intellectual and impresario
for the culture of architecture.'”
— Katharine Q. Seelye, April 17, 2022, 2:21 p.m. ET
From the cited tribute to Lippert on Holl's website —
"An excerpt from his publisher’s foreword to Anchoring :
In its iconic simplicity, his work seems to be about
the language of architecture, not in the allusive sense
used by postmodernists nor in the paradigmatic sense
used by so-called 'deconstructivists' but at the level of
essences of tropes and morphs… He is the only
American architect of his generation to be directly
influenced by the main lines in modern philosophy and
music, that is to say, by the line leading from Husserl
through to Heidegger and by separate achievements
of Bartok and Schonberg ."
Actually, although the above "iconic simplicity" passage,
up to the ellipsis after "morphs," is from the foreword
by Lippert, the references that follow the ellipsis — to
Husserl, Heidegger, Bartok, and Schonberg — are not
from Lippert's foreword, but from the introduction by
one Kenneth Frampton —
From Google Books:
Bibliographic data —
Another architectural memorial, from the reported date of Lippert's death —
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Wednesday, January 29, 2020
The 2013 Simplicity Conference…
… is reviewed by the American Mathematical Society (AMS) in
the February 2020 issue (online Jan. 27) of the AMS Notices :
See as well Simplicity Conference in this journal.
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Tuesday, May 28, 2019
“Simplification in Its Purest Form”
For students of the Hogwash School of Witchcraft and Wizardry —
"Elementary particles are the most fundamental
building blocks of nature, and their study
would seem to be an expression of simplification
in its purest form."
— Sean Carroll in The New York Times today
in an opinion piece titled "The Physicist Who
Made Sense of the Universe"
Related remarks: See a Log24 search for "Simpli…".
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Tuesday, October 23, 2018
Simplicity Versus Complexity
Simplicity (Click for some complexity.)
Complexity (Click for some simplicity.)
A passage from the 2011 book Idea Man that was suggested by
a recent New Yorker article on the book's author, the late Paul Allen —
Left-click image to enlarge.
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Sunday, October 21, 2018
Monday, August 27, 2018
Geometry and Simplicity
From …
Thinking in Four Dimensions
By Dusa McDuff
"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.
I’m a geometer and topologist, which means that
I study the structure of space …
. . . .
In each dimension there is a simplest space
called Euclidean space … "
— In Roman Kossak, ed.,
Simplicity: Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705-710, 735). Kindle Edition.
For some much simpler spaces of various
dimensions, see Galois Space in this journal.
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Sunday, December 11, 2016
Complexity to Simplicity via Hudson and Rosenhain*
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
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Complexity to Simplicity
Cézanne "showed how it was possible to pass
from the complexity of the appearance of things
to the geometrical simplicity which design demands."
— Roger Fry in the catalogue for the 1910 London
exhibition "Manet and the Post-Impressionists,"
according to …
See also A Roger Fry Reader
(edited by Christopher Reed,
University of Chicago Press, 1996).
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Tuesday, March 4, 2014
Simplify
“Alles wird viel einfacher, wenn man zuerst von der
Unendlichkeit der Theilbarkeit abstrahirt und bloss
Discrete Grössen betrachtet.”
— Carl Friedrich Gauss in 1825
(Quoted here in the July 16, 2013, post Child Buyers.)
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Saturday, July 27, 2013
Simplicity
(Continued from July 16)
From the schedule of an April 2013 philosophical conference:
Why should anyone care what Zwicky thinks?
1. Her writings. In particular, Plato as Artist .
2. Her husband. See Robert Bringhurst in this journal.
3. A reading by Zwicky and Bringhurst on March 20, 2013.
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Monday, February 14, 2011
Simplify (continued)
"Plato acknowledges how khora challenges our normal categories
of rational understanding. He suggests that we might best approach it
through a kind of dream consciousness."
—Richard Kearney, quoted here yesterday afternoon
"You make me feel like I'm living a teenage dream."
— Song at last night's Grammy awards
Richard Kiley in "Blackboard Jungle" (1955)
Note the directive on the blackboard.
Quoted here last year on this date—
Alexandre Borovik's Mathematics Under the Microscope (American Mathematical Society, 2010)—
"Once I mentioned to Gelfand that I read his Functions and Graphs ; in response, he rather sceptically asked me what I had learned from the book. He was delighted to hear my answer: 'The general principle of always looking at the simplest possible example.'….
So, let us look at the principle in more detail:
Always test a mathematical theory on the simplest possible example…
This is a banality, of course. Everyone knows it; therefore, almost no one follows it."
Related material— Geometry Simplified and A Simple Reflection Group of Order 168.
"Great indeed is the riddle of the universe.
Beautiful indeed is the source of truth."
– Shing-Tung Yau, Chairman,
Department of Mathematics, Harvard University
"Always keep a diamond in your mind."
– King Solomon at the Paradiso
Image from stoneship.org
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Sunday, May 16, 2010
Thursday, October 3, 2024
Fleetwood Thunder
"Against Dryness" —
"Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.
Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination."
— Iris Murdoch, January 1961
"the now so unfashionable naturalistic idea of character" —
"Thunder only happens when it's raining,
Players only love you when they're playing."
— Song lyric. See as well the previous post.
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Philosophy
Thesis —
A 1911 essay by T. E. Hulme,
"Romanticism and Classicism" —
"There is a general tendency to think that verse means
little else than the expression of unsatisfied emotion.
People say: 'But how can you have verse without sentiment?'
You see what it is: the prospect alarms them. A classical revival
to them would mean the prospect of an arid desert and the death
of poetry as they understand it, and could only come to fill the gap
caused by that death. Exactly why this dry classical spirit should
have a positive and legitimate necessity to express itself in poetry
is utterly inconceivable to them."
Antithesis —
A 1961 reaction against Hulme,
"Against Dryness" —
"Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.
Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination."
— Iris Murdoch, January 1961
Synthesis —
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Sunday, August 18, 2024
“Like a Kernel”
|
“The yarns of seamen have a direct simplicity, — Joseph Conrad in Heart of Darkness
“By groping toward the light we are made to realize
— Arthur Koestler, The Call Girls: A Tragi-Comedy, |
From an obituary of Alain Delon, who reportedly died today . . .
|
"He starred in the 1976 French best picture winner, 'Mr. Klein,' as a wartime German art dealer threatened by being mistaken for a Jewish man with the same name." |
See as well Felix Christian Klein in this journal.
And then there is being mistaken for a fictional archaeologist
with the same name.
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Saturday, June 15, 2024
Screwing Up a Space
Flashback to April 12, 2011 —
|
In the landscape of minimalism, John McCracken cuts a unique figure. He is often grouped with the “light and space” artists who formed the West Coast branch of the movement. Indeed, he shares interests in vivid color, new materials, and polished surfaces with fellow Californians enamored of the Kustom Kar culture. On the other hand, his signature works, the “planks” that he invented in 1966 and still makes today, have the tough simplicity and aggressive presence of New York minimalism…. “They kind of screw up a space because they lean,” McCracken has said of the planks. Their tilting, reflective surfaces activate the room, leaving the viewer uncertain of traditional boundaries. He notes that the planks bridge sculpture (identified with the floor) and painting (identified with the wall)…. His ultimate goal, as with all mystics, is unity— not just of painting and sculpture, but of substance and illusion, of matter and spirit, of art and life. Such ideas recall the utopian aspirations of early modernists like Piet Mondrian and Wassily Kandinsky. Related Art —
Unity
—Roman numeral I For a related figure, see a film review by A. O. Scott at The New York Times (September 21, 2010)— “You Will Meet a Tall Dark Stranger” begins with an unseen narrator— , sounding a lot like — paraphrasing . You may remember the quotation from high school English, about how life is a tale told by an idiot, full of sound and fury, signifying nothing. The observation is attributed to the playwright himself (“Shakespeare once said”), rather than to Macbeth, whose grim experience led him to such nihilism, but never mind. In context, it amounts to a perfectly superfluous statement of the obvious. If life signifies nothing, perhaps the tall dark figure above signifies something . Discuss. |
Related (if only phonetically) drama . . . Detective Cruz at Planck's Café.
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Friday, June 14, 2024
Summer of ’42: Python’s Monty
From April 14, 2024, some culture notes that I prefer —
An example of pi.ai's clear and pleasant conversational style
(output format simplified for presentation here) —
Can you help with computer coding problems?
Absolutely, Steven! 💻 While I can't guarantee I'll have all the answers,
I'll certainly do my best to help you debug and find solutions to your
coding conundrums. Which coding language are you working with, and what's
giving you trouble?
I want to learn how Python can permute sections of a square array.
There are several ways to permute sections of a square array in Python,
depending on your specific needs. Here are a few examples:
Using NumPy's random.permutation() function:
python
import numpy as np
arr = np.array([[1, 2], [3, 4]])
permuted_arr = np.random.permutation(arr)
Using itertools.permutations():
python
import itertools
arr = [[1, 2], [3, 4]]
permutations = list(itertools.permutations(arr))
Using custom permutation algorithms:
python
def custom_permutation(arr):
# your custom permutation logic here
arr = [[1, 2], [3, 4]]
permuted_arr = custom_permutation(arr)
Each method has its own strengths and weaknesses, and which one
you choose will depend on the size of your array, the number of
permutations you need, and your performance requirements.
I would like to use python's custom permutation method to permute
the four quadrants of a 4×4 array.
Here's an example of how to define a custom permutation function
in Python to permute the four quadrants of a 4×4 array:
python
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
In this example, we've defined a function custom_permutation()
that takes a 4×4 NumPy array as input. The function splits the array
into four quadrants (q1, q2, q3, and q4), and then uses NumPy's block()
function to create a new array with the quadrants permuted. The
resulting permuted_arr array has its quadrants rearranged.
Of course, this is just one possible permutation of the array's quadrants.
You can modify the custom_permutation() function to implement other
permutations as needed.
I tried the code you gave me and got the error “np name not defined.”
Apologies for that! I missed an import statement in the previous example.
You need to import the NumPy library at the beginning of your script to
use the np alias and its functions. Here's the corrected code:
python
import numpy as np
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
This code should work now without any issues.
It did work. Thank you, Pi.
You're very welcome, Steven! I'm glad the code worked for you.
If you have any more questions or need help with anything else,
feel free to ask.
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Sunday, April 14, 2024
Benchmarking Pi
An example of pi.ai's clear and pleasant conversational style
(output format simplified for presentation here) —
Can you help with computer coding problems?
Absolutely, Steven! 💻 While I can't guarantee I'll have all the answers,
I'll certainly do my best to help you debug and find solutions to your
coding conundrums. Which coding language are you working with, and what's
giving you trouble?
I want to learn how Python can permute sections of a square array.
There are several ways to permute sections of a square array in Python,
depending on your specific needs. Here are a few examples:
Using NumPy's random.permutation() function:
python
import numpy as np
arr = np.array([[1, 2], [3, 4]])
permuted_arr = np.random.permutation(arr)
Using itertools.permutations():
python
import itertools
arr = [[1, 2], [3, 4]]
permutations = list(itertools.permutations(arr))
Using custom permutation algorithms:
python
def custom_permutation(arr):
# your custom permutation logic here
arr = [[1, 2], [3, 4]]
permuted_arr = custom_permutation(arr)
Each method has its own strengths and weaknesses, and which one
you choose will depend on the size of your array, the number of
permutations you need, and your performance requirements.
I would like to use python's custom permutation method to permute
the four quadrants of a 4×4 array.
Here's an example of how to define a custom permutation function
in Python to permute the four quadrants of a 4×4 array:
python
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
In this example, we've defined a function custom_permutation()
that takes a 4×4 NumPy array as input. The function splits the array
into four quadrants (q1, q2, q3, and q4), and then uses NumPy's block()
function to create a new array with the quadrants permuted. The
resulting permuted_arr array has its quadrants rearranged.
Of course, this is just one possible permutation of the array's quadrants.
You can modify the custom_permutation() function to implement other
permutations as needed.
I tried the code you gave me and got the error “np name not defined.”
Apologies for that! I missed an import statement in the previous example.
You need to import the NumPy library at the beginning of your script to
use the np alias and its functions. Here's the corrected code:
python
import numpy as np
def custom_permutation(arr):
# Split array into 4 quadrants
q1 = arr[:2, :2]
q2 = arr[:2, 2:]
q3 = arr[2:, :2]
q4 = arr[2:, 2:]
# Permute quadrants
permuted_arr = np.block([
[q3, q1],
[q4, q2]
])
return permuted_arr
# Sample 4x4 array
arr = np.array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]])
permuted_arr = custom_permutation(arr)
print(permuted_arr)
This code should work now without any issues.
It did work. Thank you, Pi.
You're very welcome, Steven! I'm glad the code worked for you.
If you have any more questions or need help with anything else,
feel free to ask.
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Thursday, February 29, 2024
Masonic Mojo Dojo Casa House
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Wednesday, February 21, 2024
February Mojo
A 1911 essay by T. E. Hulme,
"Romanticism and Classicism" —
"There is a general tendency to think that verse means
little else than the expression of unsatisfied emotion.
People say: 'But how can you have verse without sentiment?'
You see what it is: the prospect alarms them. A classical revival
to them would mean the prospect of an arid desert and the death
of poetry as they understand it, and could only come to fill the gap
caused by that death. Exactly why this dry classical spirit should
have a positive and legitimate necessity to express itself in poetry
is utterly inconceivable to them."
A 1961 reaction against Hulme,
"Against Dryness" —
"Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.
Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination."
— Iris Murdoch, January 1961
Opening the Way —
For instances of character and imagination,
see this journal on February 8 and February 9.
See also the previous post and . . .
Academics may prefer "The Eureka Manifesto" —
From the MANIFESTO link in the Breakthrough Prize page above —
Our Mission . . . Should We Choose to Accept It
Comments Off on February Mojo
Tuesday, December 12, 2023
If It’s Tuesday…
|
“The yarns of seamen have a direct simplicity, — Joseph Conrad in Heart of Darkness
“By groping toward the light we are made to realize
— Arthur Koestler, The Call Girls: A Tragi-Comedy, |
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Monday, November 27, 2023
Problems with the Process
Condensed from Peter J. Cameron's weblog today —
|
“Words that tear and strange rhymes” "In his youth, Paul Simon thought of himself as a poet . . . . And surprisingly often he describes problems with the process:
For me, things were somewhat similar. Like many people, I wrote poetry in my youth. Julian Jaynes says something like 'Poems are rafts grasped at by men drowning in inadequate minds', but I think I knew from early on that one of the main reasons was to practise my writing, so that when I had something to say I could say it clearly. When Bob Dylan renounced the over-elaborate imagery of Blonde on Blonde for the clean simplicity of John Wesley Harding, I took that as a role model. Could Simon’s experience happen in mathematics? It is possible to imagine that an important mathematical truth is expressed in 'words that tear and strange rhymes'. More worryingly, an argument written in the most elegant style could be wrong, and we may be less likely to see the mistake because the writing is so good." |
The problem with the process in this case is Cameron's misheard lyrics.
From https://www.paulsimon.com/track/kathys-song-2/ —
And a song I was writing is left undone
I don’t know why I spend my time
Writing songs I can’t believe
With words that tear and strain to rhyme
A rather different artist titled a more recent song
"Strange Rhymes Can Change Minds."
See also . . .
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Thursday, November 23, 2023
Speak, Memory: 12 Panes or 16?
"Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
A differently remembered gate —
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Saturday, March 11, 2023
Wednesday, August 17, 2022
Cold Comfort Dam
"And, as with all retold tales that are in people's hearts,
there are only good and bad things and black and white
things and good and evil things and no in-between anywhere."
— John Steinbeck, author's epigraph to The Pearl
From the Season 4 finale of Westworld :
uploading Dolores's pearl at Hoover Dam —
For those who prefer greater theological simplicity . . .
Optimus Prime on a different Hoover Dam figure, that of
the AllSpark: "Before time began, there was the Cube."
Simplifying even more . . .
“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”
– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975
As above, Black's theology forms a cube.
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Monday, April 18, 2022
A Space Between
From yesterday's post "Annals of Iconic Simplicity" —
On the founding of Princeton Architectural Press:
"'There was a space between the academic,
theory-heavy M.I.T. Press and the coffeetableism
of Rizzoli,' Mr. Lamster wrote, adding that
Princeton Architectural Press would fill the gap
with 'the voice of the young practitioner.'"
Some context —
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Friday, April 1, 2022
Beauty Bare … ?
Update of 11:30 AM ET April 1, 2022 — A simpler version:
The above picture may be used to to introduce the concept of a "shape constant"
in similar figures — like the shape constant pi in a circle or the square root of 2
in a square. In each of the three similar figures at right above, the ratio of the
triangular area to the area of the attached square is a shape constant …
the same, because of their similarity, for each of the three shapes. Since the
areas of the top two triangles at right sum to that of the enclosed triangle at left,
their attached square areas sum to the area of the bottom square, Q.E.D.
The source of the proof —
Comments Off on Beauty Bare … ?
Thursday, February 17, 2022
Four Dots, Six Lines
"There is such a thing as a tesseract."
— Mrs. Whatsit in A Wrinkle in Time (1962)
"Simplify, simplify." — Henry David Thoreau in Walden (1854)
|
A Jungian on this six-line figure:
“They are the same six lines that exist in the I Ching…. Now observe the square more closely: four of the lines are of equal length, the other two are longer…. For this reason symmetry cannot be statically produced and a dance results.” |
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Tuesday, February 8, 2022
Directions Out
On reading about DNA:
"Suddenly it was clear to me
that all the beautiful complexity of life
had simplicity at its core," he says.
"This is the kind of thing mathematicians love."
— Eric Lander in "The 2004 TIME 100 — Our list
of the most influential people in the world today"
The date on the above TIME piece is Monday,
Apr. 26, 2004. Remarks in this journal on that date
are now tagged Directions Out.
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Tuesday, January 11, 2022
Architectural Metaphors
A house divided —
Update of 12:50 PM ET Tuesday, Jan. 11, 2022 —
Comments Off on Architectural Metaphors
Thursday, January 6, 2022
Sunday, January 2, 2022
Annals of Modernism: UR–Grid
The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.
Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:
See Nocciolo in this journal and . . .
Further reading on the six-set – eight-set relationship:
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Saturday, September 25, 2021
Wednesday, September 15, 2021
The Razor and the Touchstone
It is often good to remember that writers of headlines (and subheadlines)
are usually not the same people as the authors of the following texts.
In particular, in the above example, neither the word "touchstone" nor
the use of "enquires" to mean "enquiries" appears in the text proper.
Still, the mixed metaphor of "razor" as "touchstone" is not without interest.
See The Eightfold Cube and Modernist Cuts.
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Wednesday, September 8, 2021
Exalted Perception
The title is from an August 31 book review —
More recent cultural news —
"Simplify, simplify." — Thoreau.
The 4-Point Dream
::
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Monday, July 19, 2021
Seeking the Source
A post from January 21, 2003, now also tagged Big Sur,
suggested a search for the source of that post's
Shih Te poem. The result of the search —
Related Fredonia material, from posts tagged Church and Temple —
Update, from six minutes later, on related entertainment —
Update of 12:38 PM ET on the next day — July 20, 2021 —
**********************************************************************
Crystal Poem by Shide (Shih Te, etc.)
at https://terebess.hu/zen/chang/shide.html
from "Comparative List of Shide's Poems" —
Traditional Chinese:
無去無來本湛然，
不居內外及中間。
一顆水精絕瑕翳，
光明透滿出人天。
Simplified Chinese:
无去无来本湛然，
不居内外及中间。
一颗水精绝瑕翳，
光明透满出人天。
Romanization:
wú qù wú lái běn zhàn rán ，
bù jū nèi wài jí zhōng jiān 。
yī kē shuǐ jīng jué xiá yì ，
guāng míng tòu mǎn chū rén tiān 。
English Translation by Paul Rouzer:
No goings, no comings, originally tranquil;
No dwelling within or without, or at the point between.
A single crystal of purity without flaw or crack;
Its light penetrates and fills up the worlds of men and gods.
English Translation by Red Pine:
Not waxing or waning essentially still
not inside or outside and nowhere between
a single flawless crystal
whose light shines through to gods and men
*******************************************************************
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Tuesday, February 23, 2021
Wednesday, September 2, 2020
Little Metal Letters
From a report of another August 14 death —
“… on Dec. 7, 1941, ‘it seemed as though everyone at Harvard
came to the Crimson building that night, and anxiously
hung over the ticker tape [i.e., teletype ] machine to watch the
little metal letters hammer out the words that told the story.'”
— Dan Huntington Fenn Jr., quoted in his Boston Globe obituary.
“Simplicity, clarity, showing the text” — The late Howell Binkley.
“To expand the words and music and dance” . . .
See Coconut Dance.
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Thursday, May 14, 2020
Art Issue*
"… the beautiful object
that stood in
for something else.”
— Holland Cotter quoting an art historian
in The New York Times on May 13
From a post of April 27, 2020 —
“The yarns of seamen have a direct simplicity,
the whole meaning of which lies within the shell
of a cracked nut. But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode was not inside
like a kernel but outside….”
— Joseph Conrad in Heart of Darkness
The beautiful object —
Something else —
* The title is a reference to other posts now also tagged Art Issue.
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Monday, April 27, 2020
The Cracked Nut
“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”
— Thomas Wolfe, Of Time and the River
“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A8 on the octad, the kernel of this action being the translation
group of the affine space.)”
— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside…."
— Joseph Conrad in Heart of Darkness
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Sunday, April 26, 2020
Crux
This post was suggested by a David Justice weblog post yesterday,
Coincidence and Cosmos. Some related remarks —
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine.”
— Joseph Conrad in Heart of Darkness
“By groping toward the light we are made to realize
how deep the darkness is around us.”
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118
See as well posts now tagged Crux.
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Monday, October 7, 2019
Oblivion
(A sequel to Simplex Sigillum Veri and
Rabbit Hole Meets Memory Hole)
” Wittgenstein does not, however, relegate all that is not inside the bounds
of sense to oblivion. He makes a distinction between saying and showing
which is made to do additional crucial work. ‘What can be shown cannot
be said,’ that is, what cannot be formulated in sayable (sensical)
propositions can only be shown. This applies, for example, to the logical
form of the world, the pictorial form, etc., which show themselves in the
form of (contingent) propositions, in the symbolism, and in logical
propositions. Even the unsayable (metaphysical, ethical, aesthetic)
propositions of philosophy belong in this group — which Wittgenstein
finally describes as ‘things that cannot be put into words. They make
themselves manifest. They are what is mystical’ ” (Tractatus 6.522).
— Stanford Encyclopedia of Philosophy , “Ludwig Wittgenstein”
| From Tractatus Logico-Philosophicus by Ludwig Wittgenstein.
(First published in Annalen der Naturphilosophie ,1921. 5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined — a priori — to form a self-contained system. A realm subject to the law: Simplex sigillum veri. |
Somehow, the old Harvard seal, with its motto “Christo et Ecclesiae ,”
was deleted from a bookplate in an archived Harvard copy of Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906).
In accordance with Wittgenstein’s remarks above, here is a new
bookplate seal for Whitehead, based on a simplex —
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Monday, September 16, 2019
Emergence
"Elementary particles are the most fundamental building blocks
of nature, and their study would seem to be an expression of
simplification in its purest form. The essence of complexity
research, by contrast, is the emergence of new kinds of order
that are only manifest when systems are large and messy."
— Sean Carroll in an opinion piece that concludes as follows:
The above plug for Sean Carroll's book
The Big Picture : On the Origins of
Life, Meaning, and the Universe Itself suggests…
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Sunday, August 4, 2019
Bright Club
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Monday, January 7, 2019
Resonant Clarity
|
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For some context, see posts tagged Artifacts.
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Sunday, December 2, 2018
Symmetry at Hiroshima
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
|
http://www.math.sci.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt
Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve. |
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
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Friday, August 17, 2018
Hogwarts Peak
Or: Misery, Jessica … Jessica, Misery .
Related material from The Harvard Crimson —
"The beach and the castle on the hill and the waves
would always be here, always moving, always changing,
but always constant."
— Robert Miranda, "The Simplicity of Waves," August 8, 2018
Related material on waves (i.e. , "Fourier's laboratory") —
See also this journal on August 8.
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Monday, July 16, 2018
Greatly Exaggerated Report
"The novel has a parallel narrative that eventually
converges with the main story."
— Wikipedia on a book by Foer's novelist brother
Public Squares
An image from the online New York Times
on the date, July 6,
of the above Atlantic article —
An image from "Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
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Friday, July 6, 2018
Blackboard Jungle — The Prequel
An image from the online New York Times today —
Blackboard Jungle , 1955 —
"Through the unknown, remembered gate . . . ."
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Monday, July 2, 2018
Another Letterman Intro
Recognitions, Corrections; Corrections, Recognitions.
"It is the dawning of the second gestalt
in relation to the first
that is the experience of meaning."
— Jan Zwicky in "The Experience of Meaning"
(at 27:36 of 44:36 in the video of her talk)
Related remarks by the author of The Corrections —
" Even friends of The Recognitions have found it a daunting
text. Jonathan Franzen, the best known of the book's current
day champions, has offered both praise and words of warning
to potential readers. 'I loved it,' he proclaimed in the pages
of The New Yorker back in 2002, where he held up Gaddis's
novel as the preeminent example of what Franzen calls 'the
Status model' of literature. Authors who subscribe to the
'Status model' embrace fiction as the springboard for
'a discourse of genius and art-historical importance' freed
from the demands of the marketplace or the requirements of
mass consumption. Yet even Franzen acknowledges the toll
exacted by this particular masterpiece. He declares that The
Recognitions is 'the most difficult book I ever voluntarily read
in its entirety,' adding that he completed the task 'as a kind
of penance.' "
Now try Euclid.
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Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
| Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
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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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Monday, November 27, 2017
The Golay Code via Witt’s Construction
Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053.
http://scholarworks.sjsu.edu/etd_theses/4053.
See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.
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Tuesday, October 10, 2017
Dueling Formulas
Continued from the previous post and from posts
now tagged Dueling Formulas —
The four-diamond formula of Jung and
the four-dot "as" of Claude Lévi-Strauss:
Simplified versions of the diamonds and the dots —
::
I prefer Jung. For those who prefer Lévi-Strauss —
First edition, Cornell University Press, 1970.
A related tale — "A Meaning, Like."
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Thursday, May 18, 2017
Bit Plot
From a May 15 review of a new book by Douglas Coupland, author of
the 1991 book Generation X: Tales for an Accelerated Culture .
"Minimalists are actually extreme hoarders:
they hoard space." — Douglas Coupland
The title of Coupland's new book suggests a review of Schmeikal
in this journal …
Coupland's above remark on hoarders suggests a look at
a wealthy California collector whom, were he not wealthy,
some might call a hoarder.
“I buy things because they strike an emotional bell,
they appeal to my curiosity, to the thrill of discovery
of the extraordinary in the ordinary,” Mr. Cotsen told
The Denver Post in 1998. “They appeal to my sense
of humor, and to my search for the beauty in simplicity.”
He added, “I decided I had a collection when there was
no more space to put anything.”
By the time he died at 88 on May 8 in Beverly Hills, Calif.,
Mr. Cotsen (pronounced COAT-zen) had donated about
half of the material in his collections to institutions like the
Asian Art Museum in San Francisco, Princeton University
and the Museum of International Folk Art in Santa Fe, N.M.
— Richard Sandomir in the online New York Times , May 17
Cotsen reportedly died at 88 on May 8.
See also this journal on that date —
|
Monday, May 8, 2017
New Pinterest Board
|
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Friday, April 7, 2017
Personal Identity
From "The Most Notorious Section Phrases," by Sophie G. Garrett
in The Harvard Crimson on April 5, 2017 —
This passage reminds me of (insert impressive philosophy
that was not in the reading).
This student is just being a show off. We get that they are smart
and well read. Congrats, but please don’t make the rest of the us
look bad in comparison. It should be enough to do the assigned
reading without making connections to Hume’s theory of the self.
Hume on personal identity (the "self") —
|
For my part, when I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold, light or shade, love or hatred, pain or pleasure. I never can catch myself at any time without a perception, and never can observe any thing but the perception. When my perceptions are removed for any time, as by sound sleep, so long am I insensible of myself, and may truly be said not to exist. And were all my perceptions removed by death, and could I neither think, nor feel, nor see, nor love, nor hate, after the dissolution of my body, I should be entirely annihilated, nor do I conceive what is further requisite to make me a perfect nonentity. I may venture to affirm of the rest of mankind, that they are nothing but a bundle or collection of different perceptions, which succeed each other with an inconceivable rapidity, and are in a perpetual flux and movement. Our eyes cannot turn in their sockets without varying our perceptions. Our thought is still more variable than our sight; and all our other senses and faculties contribute to this change: nor is there any single power of the soul, which remains unalterably the same, perhaps for one moment. 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. The comparison of the theatre must not mislead us. They are the successive perceptions only, that constitute the mind; nor have we the most distant notion of the place where these scenes are represented, or of the materials of which it is composed. |
Related material —
Imago Dei in this journal.
Backstory —
The previous post
and The Crimson Abyss.
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Monday, April 11, 2016
Princeton Style
The previous post contained a remark from Princeton
on the January 1, 2016, death of a mathematician.
"There's a certain Princeton style that focuses on
precision, centrality and simplicity."
See also …
For a different sort of style, see Death on New Year's Day.
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Gospel of the Nobodies
"Principles before personalities" — AA saying
Principles
From an April 8 Princeton obituary of a mathematician —
" Moore embodied a 'Princeton style' that made him
a challenging and influential presence in the careers
of his students, said Joseph Neisendorfer, a professor
of mathematics at the University of Rochester who
received his Ph.D. in mathematics from Princeton in
1972. Because of Moore's style, his students would
write theses that 'almost without exception' were
significant advances in mathematics, Neisendorfer said.
'There's a certain Princeton style that focuses on
precision, centrality and simplicity. He was a superb
mathematician and he exercised a lot of influence
by imparting his style to his students,' Neisendorfer said.
'He epitomized the Princeton style.' "
Personalities
Gospel of the Nobodies
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Sunday, April 10, 2016
Synchronicity Check
The book quoted in the previous post, Attack of the Copula Spiders,
was reportedly published on March 27, 2012.
For the Church of Synchronology —
The above icon may be viewed as a simplified version
of the image described in the April 8 post Space Cross.
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Tuesday, February 23, 2016
Both Sides Now
(Continued from the link in the previous post to
a Feb. 20 NY Times essay on the brain's two sides)
From a webpage on Galois geometry —
|
Postscript From a 2002 review by Stacy G. Langton of Sherman Stein's book on mathematics, How the Other Half Thinks : "The title of Stein's book (perhaps chosen by the publisher?) seems to refer to the popular left brain/right brain dichotomy. As Stein writes (p. ix): 'I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive, holistic) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role.' Stein does well to avoid identifying mathematics with the activity of just one side of the brain. He would have done better, however, not to have endorsed the left brain/right brain ideology. While it does indeed appear to be the case that the two sides of our brain act in rather different ways, the idea that the right brain is 'intuitive, holistic,' while the left brain is 'analytical, numerical,' is a vast oversimplification, and goes far beyond the actual evidence." |
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Thursday, November 26, 2015
Charm School
"When the first Harry Potter book appeared, in 1997,
it was just a year before the universal search engine
Google was launched. And so Hermione Granger,
that charming grind, still goes to the Hogwarts library
and spends hours and hours working her way through
the stacks, finding out what a basilisk is or how to
make a love potion."
— Adam Gopnik in The New Yorker issue dated
St. Valentine's Day, 2011
More recently, Gopnik writes that …
"Arguing about non-locality went out of fashion, in this
account, almost the way 'Rock Around the Clock'
displaced Sinatra from the top of the charts."
— Adam Gopnik in The New Yorker issue dated
St. Andrew's Day, 2015
This journal on Valentine's Day, 2011 —
"One heart will wear a valentine." — Sinatra
"… she has written a love letter to Plato, whom
she regards as having given us philosophy.
He is, in her view, as relevant today as he ever
was — which is to say, very."
— New York Times review of a book by
Rebecca Newberger Goldstein, April 18, 2014
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Tuesday, May 12, 2015
Zinsser Obituary
"William Zinsser, a writer, editor and teacher
whose book ‘On Writing Well’ sold more than
1.5 million copies by employing his own literary
craftsmanship to urge clarity, simplicity, brevity
and humanity, died on Tuesday [May 12, 2015]
at his home in Manhattan. He was 92."
— Douglas Martin in the online New York Times
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Saturday, March 14, 2015
Unicode Diamonds
The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.
◢◣
◥◤
Some variations:
◤◥
◣◢
◤◥
◢◣
◤◣
◢◥
◤◣
◥◢
Such combined Unicode characters —
◢ black lower right triangle,
◣ black lower left triangle,
᭘ black upper left triangle,
᭙ black upper right triangle
— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.
The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.
Update of 11:20 PM ET March 15, 2015 —
Seekers of simplicity should note that there is
a simple program in the Processing.js language, not using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.
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Sunday, September 14, 2014
Sermon
Epigraphs from Parallelisms of Complete Designs
by Peter J. Cameron (Cambridge University Press, 1976)
Introduction
Through the unknown, remembered gate
When the last of earth left to discover
Is that which was the beginning
(T. S. Eliot: Little Gidding)
I The existence theorem
Here the impossible union
Of spheres of existence is actual
(T. S. Eliot: The Dry Salvages)
II The parallelogram property
A condition of complete simplicity
(Costing not less than everything)
(T. S. Eliot: Little Gidding)
III Steiner points and Veblen points
You say I am repeating
Something I have said before. I shall say it again.
Shall I say it again?
(T. S. Eliot: East Coker)
IV Edge-colourings of complete graphs
And hollyhocks that aim too high
Red into grey and tumble down
(T. S. Eliot: East Coker)
V Biplanes and metric regularity
Two and two, necessarye conjunction,
Holding eche other by the hand or the arm
Whiche betokeneth concorde.
(T. S. Eliot: East Coker)
VI Automorphism groups
At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement.
(T. S. Eliot: Burnt Norton)
VII Resolutions and partition systems
… fiddle with pentagrams
Or barbituric acids, or dissect
The recurrent image into pre-conscious terrors .. .
(T. S. Eliot: The Dry Salvages)
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Thursday, August 7, 2014
Abuse of Language
From Wikipedia — Abuse of language —
“… in mathematics, a use of terminology in a way that is not formally correct
but that simplifies exposition or suggests the correct intuition.”
The phrase “symplectic structure” in the previous post
was a deliberate abuse of language. The real definition:
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Friday, July 25, 2014
Magic in the Moonshine
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine.”
— Joseph Conrad in Heart of Darkness
“By groping toward the light we are made to realize
how deep the darkness is around us.”
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118
“Spectral evidence is a form of evidence
based upon dreams and visions.” —Wikipedia
See also Moonshine (May 15, 2014) and, from the date of the above
New York Times item, two posts tagged Wunderkammer .
Related material: From the Spectrum program of the Mathematical
Association of America, some non-spectral evidence.
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Thursday, May 15, 2014
Moonshine
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine.”
— Joseph Conrad in Heart of Darkness
Photo of full moon over Oslo last night by Josefine Lyche:
A scene from my film viewing last night:
Some background (click to enlarge):
Note:
The “I, Frankenstein” scene above should not be interpreted as
a carrying of Martin Gardner through a lyche gate. Gardner
is, rather, symbolized by the asterisk in the first image from
the above Google search.
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Friday, April 4, 2014
Eight Gate
From a Huffington Post discussion of aesthetics by Colm Mulcahy
of Spelman College, Atlanta:
"The image below on the left… is… overly simplistic, and lacks reality:
It's all a matter of perspective: the problem here is that opposite sides
of the cube, which are parallel in real life, actually look parallel in the
left image! The image on the right is better…."
A related discussion: Eight is a Gate.
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Friday, March 14, 2014
Wednesday, November 13, 2013
Rated X
New York Times theater critic Ben Brantley
last night at 10 PM ET on the opening of a
play by Samuel Beckett —
"The cause of this incontinent mirth?
The dirtiest joke of all time. I mean life itself.
No playwright of the 20th century, and quite
possibly ever, has told this joke with the
clarity, simplicity and richness of Beckett."
Related material — This journal yesterday.
See also Lead Balloon.
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Thursday, September 5, 2013
Kernel and Glow
"The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine."
— Joseph Conrad in Heart of Darkness
Kernel — See Nocciolo.
Glow — See Moonshine and Moonshine II.
See also Cold Open (Jan. 29, 2011) and
Where Entertainment is God (Aug. 25, 2013).
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Sunday, July 28, 2013
Sermon
(Simplicity continued)
"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by the
demonstration or the metaphor.
What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already-
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."
— Jan Zwicky, Wisdom & Metaphor , page 36 (left)
Zwicky illustrates this with Plato's diamond figure
from the Meno on the facing page— her page 36 (right).
A more sophisticated geometrical figure—
Galois-geometry key to
Desargues' theorem:
| D | E | F | |
| S' | P | Q | R |
| S | P' | Q' | R' |
| O | P1 | Q1 | R1 |
For an explanation, see
Classical Geometry in Light of Galois Geometry.
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Tuesday, July 16, 2013
Child Buyers
The title refers to a classic 1960 novel by John Hersey.
“How do you get young people excited about space?”
— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)
Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society—
“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”
This suggests a related image by Gauss, with a
remark on simplicity—
Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).
Related material that does feature the somewhat simpler 3×3 array
of squares, not seen as part of an infinite array—
Marketing the Holy Field
Click image for the original post.
For a purely mathematical view of the holy field, see Visualizing GL(2,p).
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Saturday, May 4, 2013
Structure vs. Character
"… Reality is not a given whole. An understanding of this,
a respect for the contingent, is essential to imagination
as opposed to fantasy. Our sense of form, which is an
aspect of our desire for consolation, can be a danger to
our sense of reality as a rich receding background.
Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.
Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination.
Think of the Russians, those great masters of the contingent.
Too much contingency of course may turn art into journalism.
But since reality is incomplete, art must not be too much
afraid of incompleteness. Literature must always represent a
battle between real people and images; and what it requires
now is a much stronger and more complex conception of the
former."
— Iris Murdoch, January 1961, "Against Dryness"
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Sunday, March 10, 2013
Galois Space
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
- Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).For the first two pages, click here.
- Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
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Wednesday, February 13, 2013
Form:
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M24. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
Related material:
|
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) |
Those who prefer story to structure may consult
- today's previous post on the Penrose diamond
- the remarks of Scott Aaronson on August 17, 2012
- the remarks in this journal on that same date
- the geometry of the 4×4 array in the context of M24.
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Wednesday, October 24, 2012
In a Nutshell
"The yarns of seamen have a direct simplicity,
the whole meaning of which
lies within the shell of a cracked nut.
But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode
was not inside like a kernel but outside,
enveloping the tale which brought it out
only as a glow brings out a haze,
in the likeness of one of these misty halos
that sometimes are made visible by
the spectral illumination of moonshine."
— Joseph Conrad in Heart of Darkness
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Sunday, April 22, 2012
Plan 9 from MIT
Click image for some context from MIT.
Background from The New York Times— "Can You Make Yourself Smarter?"
See also "Plan 9" in this journal.
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Friday, April 20, 2012
Complex Reflection
Yesterday's post in memory of Octavio Paz—
|
… the free-standing, two-sided “Life-Death Figure,” |
An earlier post yesterday, Fashion Notes, linked to a Sting video—
|
From "Loo Ree," by Zenna Henderson "It's so hard to explain–" "Oh, foof!" I cried defiantly, taking off my glasses and, smearing the tears across both lenses with a tattered Kleenex. "So I'm a dope, a moron! If I can explain protective coloration to my six-year-olds and the interdependence of man and animals, you can tell me something of what the score is!" I scrubbed the back of my hand across my blurry eyes. "If you have to, start out 'Once upon a time."' I sat down– hard. Loo Ree smiled and sat down, too. "Don't cry, teacher. Teachers aren't supposed to have tears." "I know it," I sniffed. "A little less than human-that's us." "A little more than human, sometimes." Loo Ree corrected gently. "Well then, you must understand that I'll have to simplify. You will have to dress the bare bones of the explanation according to your capabilities. "Once upon a time there was a classroom. Oh, cosmic in size, but so like yours that you would smile in recognition if you could see it all. And somewhere in the classroom something was wrong. Not the whispering and murmuring– that's usual. Not the pinching and poking and tattling that goes on until you get so you don't even hear it." I nodded. How well I knew. "It wasn't even the sudden blow across the aisle or the unexpected wrestling match in the back of the room. That happens often, too. But something else was wrong. It was an undercurrent, a stealthy, sly sort of thing that has to be caught early or it disrupts the whole classroom and tarnishes the children with a darkness that will never quite rub off. "The teacher could feel it –as all good teachers can– and she spoke to the principal. He, being a good principal, immediately saw the urgency of the matter and also saw that it was beyond him, so he called in an Expert." "You?" I asked, feeling quite bright because I had followed the analogy so far. Loo Ree smiled. "Well, I'm part of the Expert." |
"If you have to, start out 'Once upon a time.'"
Yesterday's Paz post was at 6:48 PM EDT.
For the autistic, here is some related mathematics.
Yesterday's Fashion Notes post was at 1:06 PM EDT.
A related chronological note from Rolling Stone yesterday—
"Levon Helm, singer and drummer for the Band,
died on April 19th in New York of throat cancer.
He was 71.
"He passed away peacefully at 1:30 this afternoon…."
Helm and The Band performing "The Weight"—
"I pulled into Nazareth, I was a-feelin' 'bout half past dead…"
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Tuesday, October 11, 2011
24 Hour DeLillo
Review of DeLillo's novel Point Omega—
"One difference between art and entertainment has to do with the speed of perception. Art deliberately slows and complicates reading, hearing, and/or viewing so that you’re challenged to re-think and re-feel form and experience. Entertainment deliberately accelerates and simplifies them so that you don’t have to think about or feel very much of anything at all except, perhaps, the adrenalin rush before dazzling spectacle. Although, of course, there can be myriad gradations between the former and latter, in their starkest articulation we’re talking about the distance between, say, David Foster Wallace’s Infinite Jest and Dan Brown’s The Lost Symbol…."
— Lance Olsen, March 1, 2010, in The Quarterly Conversation
Robert Hughes on fast and slow art—
"We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn't merely sensational, that doesn't get its message across in 10 seconds, that isn't falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game."
– Speech of June 1, 2004
Log24 on art speeds—
A Study in Art Education (June 15, 2007)
Twenty-four (March 13, 2011)
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Wednesday, July 20, 2011
Cover Art
The Misalignment of Mars and Venus
A death in Sarasota on Sunday leads to a weblog post from Tuesday
that suggests a review of Dan Brown's graphic philosophy—
|
From The Da Vinci Code : Langdon pulled a pen from his pocket. “Sophie are you familiar with the modern icons for male and female?” He drew the common male symbol ♂ and female symbol ♀. “Of course,” she said. “These,” he said quietly, are not the original symbols for male and female. Many people incorrectly assume the male symbol is derived from a shield and spear, while the female represents a mirror reflecting beauty. In fact, the symbols originated as ancient astronomical symbols for the planet-god Mars and the planet-goddess Venus. The original symbols are far simpler.” Langdon drew another icon on the paper. ∧
“This symbol is the original icon for male ,” he told her. “A rudimentary phallus.” “Quite to the point,” Sophie said. “As it were,” Teabing added. Langdon went on. “This icon is formally known as the blade , and it represents aggression and manhood. In fact, this exact phallus symbol is still used today on modern military uniforms to denote rank.” “Indeed.” Teabing grinned. “The more penises you have, the higher your rank. Boys will be boys.”
Langdon winced. “Moving on, the female symbol, as you might imagine, is the exact opposite.” He drew another symbol on the page. “This is called the ∨ Sophie glanced up, looking surprised. Langdon could see she had made the connection. “The chalice,” he said, “resembles a cup or vessel, and more important, it resembles the shape of a woman’s womb. This symbol communicates femininity, womanhood, and fertility.” |
Langdon's simplified symbols, in disguised form, illustrate
a musical meditation on the misalignment of Mars and Venus—
This was adapted from an album cover by "Meyers/Monogram"—
See also Secret History and The Story of N.
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Tuesday, May 10, 2011
Groups Acting
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).
A high-minded— if not Shakespearean— non-Nordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394
"Groups act "… For some examples, see
- The 2×2×2 Cube,
- The Diamond 16 Puzzle,
- The Diamond Theorem, and
- Finite Geometry of the Square and Cube.
Related entertainment—
High-minded— Many Dimensions—
Not so high-minded— The Cosmic Cube—
One way of blending high and low—
The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordic-averse, some background on the Hebrew connection—
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Thursday, May 5, 2011
On Art and Magic
Two Blocks Short of a Design:
A sequel to this morning’s post on Douglas Hofstadter
Photo of Hofstadter by Mike McGrath taken May 13, 2006 |
Related material — See Lyche’s “Theme and Variations” in this journal
and Hofstadter’s “Variations on a Theme as the Essence of Imagination”
— Scientific American October 1982
A quotation from a 1985 book by Hofstadter—
“… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.
Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas . (It is not for nothing that the word
‘magic’ appears inside the title!)”
The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.
An example from Wikipedia —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
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Tuesday, April 12, 2011
Unique Figure
In the landscape of minimalism, John McCracken cuts a unique figure. He is often grouped with the “light and space” artists who formed the West Coast branch of the movement. Indeed, he shares interests in vivid color, new materials, and polished surfaces with fellow Californians enamored of the Kustom Kar culture. On the other hand, his signature works, the “planks” that he invented in 1966 and still makes today, have the tough simplicity and aggressive presence of New York minimalism….
“They kind of screw up a space because they lean,” McCracken has said of the planks. Their tilting, reflective surfaces activate the room, leaving the viewer uncertain of traditional boundaries. He notes that the planks bridge sculpture (identified with the floor) and painting (identified with the wall)….
His ultimate goal, as with all mystics, is unity— not just of painting and sculpture, but of substance and illusion, of matter and spirit, of art and life. Such ideas recall the utopian aspirations of early modernists like Piet Mondrian and Wassily Kandinsky.
Related Art —
Unity
—Roman numeral I
as well as capital I
For a related figure, see a film review by A. O. Scott at The New York Times (September 21, 2010)—
“You Will Meet a Tall Dark Stranger” begins with an unseen narrator— , sounding a lot like — paraphrasing . You may remember the quotation from high school English, about how life is a tale told by an idiot, full of sound and fury, signifying nothing. The observation is attributed to the playwright himself (“Shakespeare once said”), rather than to Macbeth, whose grim experience led him to such nihilism, but never mind. In context, it amounts to a perfectly superfluous statement of the obvious.
If life signifies nothing, perhaps the tall dark figure above signifies something . Discuss.
Related Art Criticism —
For more on light and space, see this journal on the date of McCracken’s death—
Note planks.
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Sunday, April 3, 2011
On to Chicago!
Commentary on last night—
Tonight: The After-Party.
In related news…
"The yarns of seamen have a direct simplicity, the whole meaning of which
lies within the shell of a cracked nut. But Marlow was not typical
(if his propensity to spin yarns be excepted), and to him the meaning
of an episode was not inside like a kernel but outside, enveloping the tale
which brought it out only as a glow brings out a haze, in the likeness of
one of these misty halos that sometimes are made visible by
the spectral illumination of moonshine."
– Joseph Conrad in Heart of Darkness , quoted here in
Cold Open (Saturday night, January 29, 2011)
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Friday, March 18, 2011
Defining Configurations*
The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.
From that article:
- DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
- EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
The following corrects the word "unique" in the example.
* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.
_____________________________________________________________
Update of March 21
The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—
In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.
Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.
Update of March 22 (itself updated on March 25)
For further background on configurations, see Dolgachev—
Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.
Update of March 27
See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.
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Saturday, January 29, 2011
Cold Open
"The yarns of seamen have a direct simplicity, the whole meaning of which lies within the shell of a cracked nut. But Marlow was not typical (if his propensity to spin yarns be excepted), and to him the meaning of an episode was not inside like a kernel but outside, enveloping the tale which brought it out only as a glow brings out a haze, in the likeness of one of these misty halos that sometimes are made visible by the spectral illumination of moonshine."
— Joseph Conrad in Heart of Darkness
Some background—
An image from yesterday's search
God, TIme, Hopkins
"We got tom-toms over here bigger than a monster
Bla Bla Bla Bla Bla Bla Bla Bla"
— "Massive Attack"
"I'm just checking your math on that. Yes, I got the same thing."
"Live… Uh, check that… From New York, it's Saturday Night! "
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Thursday, December 2, 2010
Caesarian
The Dreidel Is Cast
The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."
That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.
The holiday is celebrated with, among other things, the Jewish version of a die— the dreidel . Note the similarity of the dreidel to an illustration of The Stone* on the cover of the 2001 Eerdmans edition of Charles Williams's 1931 novel Many Dimensions—
For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.
Here is the opening of Many Dimensions—
For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.
Some further holiday meditations on a beginning—
Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography—
"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-
those branches of mathematics in which I had been personally involved.
He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-
The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."
For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.
The following remarks by Hyland are of more general interest—
"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."
— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.
Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota—
See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.
* Williams's novel says the letters of The Stone are those of the Tetragrammaton— i.e., Yod, He, Vau, He (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone include the three-pronged letter Shin , also found on the dreidel . What esoteric religious meaning is implied by this, I do not know.
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Thursday, September 16, 2010
For the Pope in Scotland
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Saturday, August 7, 2010
The Matrix Reloaded
For aficionados of mathematics and narrative —
Illustration from
"The Galois Quaternion— A Story"
This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean plane, but rather with unit squares
representing points in a finite Galois affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.
See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.
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Wednesday, July 28, 2010
Without Diamond-Blazons
Excerpt from Wallace Stevens's
"The Pediment of Appearance"—
Young men go walking in the woods,
Hunting for the great ornament,
The pediment* of appearance.
They hunt for a form which by its form alone,
Without diamond—blazons or flashing or
Chains of circumstance,
By its form alone, by being right,
By being high, is the stone
For which they are looking:
The savage transparence.
* Pediments, triangular and curved—
— From "Stones and Their Stories," an article written
and illustrated by E.M. Barlow, copyright 1913.
Related geometry—
(See Štefan Porubský: Pythagorean Theorem .)
A proof with diamond-blazons—
(See Ivars Peterson's "Square of the Hypotenuse," Nov. 27, 2000.)
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Thursday, July 15, 2010
Brightness at Noon, continued
"What exactly was Point Omega?"
This is Robert Wright in Nonzero: The Logic of Human Destiny.
Wright is discussing not the novel Point Omega by Don DeLillo,
but rather a (related) concept of the Jesuit philosopher Pierre Teilhard de Chardin.
My own idiosyncratic version of a personal "point omega"—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
For some other sorts of points, see the drawings
on the wall and Geometry Simplified—
The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affine-space squares.
For related darkness at noon, see Derrida on différance
as a version of Plato's khôra—
The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institution— Deconstruction in a Nutshell—
For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.
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Monday, July 12, 2010
Through Phenomenology to Thought
![Image-- 'Heidegger: Through Phenomenology to Thought,' page 501, on a 'simple middle [-point]'](http://www.log24.com/log/pix10A/100712-MiddlePoint.jpg)
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Tuesday, July 6, 2010
Thoreau on Group Theory
"Instead of a million count half a dozen." —Walden
"Of all the symmetric groups, S6 is perhaps the most remarkable."
— Notes 2 (Autumn 2008), apparently by Robert A. Wilson,
for Group Theory, MTH714U
For a connection of MTH714U with Walden, see "Window, continued."
For a connection of "Window" with the remarkable S6, see Inscapes.
For some deeper background, see Wilson's "Exceptional Simplicity."
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Window, continued
| “Simplicity, simplicity, simplicity! I say, let your affairs be as two or three, and not a hundred or a thousand; instead of a million count half a dozen, and keep your accounts on your thumb-nail.” — Henry David Thoreau, Walden This quotation is the epigraph to |
From Peter J. Cameron’s review notes for
his new course in group theory—
From Log24 on June 24—
Geometry Simplified
(an affine space with subsquares as points
and sets of subsquares as hyperplanes)
(a projective space with, as points, sets
of line segments that separate subsquares)
Exercise—
Show that the above geometry is a model
for the algebra discussed by Cameron.
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Thursday, June 24, 2010
Midsummer Noon
Geometry Simplified
(a projective space)
The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)
The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).
Homogeneous coordinates for the
points of this line —
(1,0), (0,1), (1,1).
Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).
The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —
(an affine space)
The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.
For further details, see Galois Geometry.
There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —
Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.
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Friday, June 11, 2010
Toward the Light
The title is a reference to yesterday's noon post.
For the late Vladimir Igorevich Arnold—
"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical Mathematicks of the City of Heaven."
— Sir Thomas Browne, The Garden of Cyrus, Chapter V
Arnold's own mystical mathematics may be found in his paper
"Polymathematics: Is Mathematics a Single Science or a Set of Arts?"
Page 13–
"In mathematics we always encounter mysterious analogies, and our trinities [page 8] represent only a small part of these miracles."
Also from that paper—
Page 5, footnote 2–
"The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization."
Arnold died in Paris on June 3. A farewell gathering was held there on June 8—
"Celles et ceux qui le souhaitent pourront donner un dernier adieu à Vladimir Igorevitch
mardi 8 juin, de 14h a 16h, chambre mortuaire de l'hopital Saint Antoine…."
An International Blue Diamond
In Arnold's memory— Here, in the Russian style, is a link to a "first nontrivial case" of a blue diamond— from this journal on June 8 (feast of St. Gerard Manley Hopkins). For those who prefer French style, here is a link to a blue diamond from May 18—
From French cinema—
"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa"
Comments Off on Toward the Light
Monday, April 26, 2010
Types of Ambiguity
From Ursula K. Le Guin’s novel
The Dispossessed: An Ambiguous Utopia (1974)—
Chapter One
“There was a wall. It did not look important. It was built of uncut rocks roughly mortared. An adult could look right over it, and even a child could climb it. Where it crossed the roadway, instead of having a gate it degenerated into mere geometry, a line, an idea of boundary. But the idea was real. It was important. For seven generations there had been nothing in the world more important than that wall.
Like all walls it was ambiguous, two-faced. What was inside it and what was outside it depended upon which side of it you were on.”
Note—
“We note that the phrase ‘instead of having a gate it degenerated into mere geometry’ is mere fatuousness. If there is an idea here, degenerate, mere, and geometry in concert do not fix it. They bat at it like a kitten at a piece of loose thread.”
— Samuel R. Delany, The Jewel-Hinged Jaw: Notes on the Language of Science Fiction (Dragon Press, 1977), page 110 of revised edition, Wesleyan University Press, 2009
(For the phrase mere geometry elsewhere, see a note of April 22. The apparently flat figures in that note’s illustration “Galois Affine Geometry” may be regarded as degenerate views of cubes.)
Later in the Le Guin novel—
“… The Terrans had been intellectual imperialists, jealous wall builders. Even Ainsetain, the originator of the theory, had felt compelled to give warning that his physics embraced no mode but the physical and should not be taken as implying the metaphysical, the philosophical, or the ethical. Which, of course, was superficially true; and yet he had used number, the bridge between the rational and the perceived, between psyche and matter, ‘Number the Indisputable,’ as the ancient founders of the Noble Science had called it. To employ mathematics in this sense was to employ the mode that preceded and led to all other modes. Ainsetain had known that; with endearing caution he had admitted that he believed his physics did, indeed, describe reality.
Strangeness and familiarity: in every movement of the Terran’s thought Shevek caught this combination, was constantly intrigued. And sympathetic: for Ainsetain, too, had been after a unifying field theory. Having explained the force of gravity as a function of the geometry of spacetime, he had sought to extend the synthesis to include electromagnetic forces. He had not succeeded. Even during his lifetime, and for many decades after his death, the physicists of his own world had turned away from his effort and its failure, pursuing the magnificent incoherences of quantum theory with its high technological yields, at last concentrating on the technological mode so exclusively as to arrive at a dead end, a catastrophic failure of imagination. Yet their original intuition had been sound: at the point where they had been, progress had lain in the indeterminacy which old Ainsetain had refused to accept. And his refusal had been equally correct– in the long run. Only he had lacked the tools to prove it– the Saeba variables and the theories of infinite velocity and complex cause. His unified field existed, in Cetian physics, but it existed on terms which he might not have been willing to accept; for the velocity of light as a limiting factor had been essential to his great theories. Both his Theories of Relativity were as beautiful, as valid, and as useful as ever after these centuries, and yet both depended upon a hypothesis that could not be proved true and that could be and had been proved, in certain circumstances, false.
But was not a theory of which all the elements were provably true a simple tautology? In the region of the unprovable, or even the disprovable, lay the only chance for breaking out of the circle and going ahead.
In which case, did the unprovability of the hypothesis of real coexistence– the problem which Shevek had been pounding his head against desperately for these last three days. and indeed these last ten years– really matter?
He had been groping and grabbing after certainty, as if it were something he could possess. He had been demanding a security, a guarantee, which is not granted, and which, if granted, would become a prison. By simply assuming the validity of real coexistence he was left free to use the lovely geometries of relativity; and then it would be possible to go ahead. The next step was perfectly clear. The coexistence of succession could be handled by a Saeban transformation series; thus approached, successivity and presence offered no antithesis at all. The fundamental unity of the Sequency and Simultaneity points of view became plain; the concept of interval served to connect the static and the dynamic aspect of the universe. How could he have stared at reality for ten years and not seen it? There would be no trouble at all in going on. Indeed he had already gone on. He was there. He saw all that was to come in this first, seemingly casual glimpse of the method, given him by his understanding of a failure in the distant past. The wall was down. The vision was both clear and whole. What he saw was simple, simpler than anything else. It was simplicity: and contained in it all complexity, all promise. It was revelation. It was the way clear, the way home, the light.”
Related material—
Time Fold, Halloween 2005, and May and Zan.
See also The Devil and Wallace Stevens—
“In a letter to Harriet Monroe, written December 23, 1926, Stevens refers to the Sapphic fragment that invokes the genius of evening: ‘Evening star that bringest back all that lightsome Dawn hath scattered afar, thou bringest the sheep, thou bringest the goat, thou bringest the child home to the mother.’ Christmas, writes Stevens, ‘is like Sappho’s evening: it brings us all home to the fold’ (Letters of Wallace Stevens, 248).”
— “The Archangel of Evening,” Chapter 5 of Wallace Stevens: The Intensest Rendezvous, by Barbara M. Fisher, The University Press of Virginia, 1990
Comments Off on Types of Ambiguity
Sunday, April 25, 2010
Sunday, February 14, 2010
Sunday School
"Simplify, simplify." — Henry David Thoreau
"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."
— Alexandre Borovik, 2010 (See previous entry.)
Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page
A Simple Reflection Group
of Order 168.
Background:
Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups—
15.1 (p. 114): Every finite reflection group is a Coxeter group.
15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.
Consider in this context the above simple reflection group of order 168.
(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)
Comments Off on Sunday School
Example
From Alexandre Borovik's new book
Mathematics Under the Microscope
(American Mathematical Society, 2010)–
Related material:
Finite Geometry and Physical Space
(Good Friday, 2009)
This kindergarten-level discussion of
the simple group of order 168
also illustrates Thoreau's advice:
"Simplicity, simplicity, simplicity!"
Comments Off on Example
Sunday, May 17, 2009
Sunday May 17, 2009
Design Theory
Laura A. Smit, Calvin College, "Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams"–
"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."
Hans Ludwig de Vries, "On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16)," Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–
"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."
1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).
4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).
Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–
"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."
"Simplify, simplify."
— Thoreau
"Beauty is bound up
with symmetry."
— Weyl
Pottonen's thesis is
dated Nov. 16, 2005.
For some remarks on
images and theology,
see Log24 on that date.
Click on the above image
for some further details.
Comments Off on Sunday May 17, 2009
Friday, April 10, 2009
Friday April 10, 2009
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The Non-Euclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
Comments Off on Friday April 10, 2009
Tuesday, January 13, 2009
Tuesday January 13, 2009
Something Traditional —
“German Chancellor Dr. Angela Merkel is the Charlemagne Prize laureate of 2008…. The prize will be awarded on 1 May, Ascension Day.”
Something Modern —
Previously undescribed in this journal:
| A NOTE BY THE DESIGNER
“The traditional chess set, with its naturalistic images of medieval armies, suggests a game between combatants who enjoy the winning of battles. This chess set, with its articulated images of abstract force, suggests a game between contestants who enjoy the process of thinking.
The primary principle of this design… is that the operating reality or function of each piece– both its value and how it moves– is embodied in a simple self-expressive form….  Design Copyright F. Lanier Graham 1967
— F. Lanier Graham, 1967 |
For those whose tastes in recreation are less rational, there is also the legendary chess set of Charlemagne described in novels by Katherine Neville. (See ART WARS.)
Related material: this journal on the First of May, 2008, the date of last year’s Charlemagne award.
Comments Off on Tuesday January 13, 2009
Friday, December 19, 2008
Friday December 19, 2008
Inside the
White Cube
White Cube
Part I: The White Cube
Part II: Inside
Part III: Outside
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
|
From Chapter 3,
"Sutures* of Structures," p. 58: "What, then, is a frame, and what is frame work? This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'" * P. 61:
|
Comments Off on Friday December 19, 2008
Sunday, November 16, 2008
Sunday November 16, 2008
ART WARS
continued
continued
From Koestler’s Darkness at Noon, a fictional Communist on propaganda:
“It is necessary to hammer every sentence into the masses by repetition and simplification. What is presented as right must shine like gold; what is presented as wrong must be black as pitch.”
Thanks for this quotation to Kati Marton, author of The Great Escape: Nine Jews Who Fled Hitler and Changed the World (Simon & Schuster, paperback edition Nov. 6, 2007). One of Marton’s nine was Koestler.
From another book related to this exodus:
“Riesz was one of the most elegant mathematical writers in the world, known for his precise, concise, and clear expositions. He was one of the originators of the theory of function spaces– an analysis which is geometrical in nature.”
— Stanislaw Ulam, Adventures of a Mathematician
And from Gian-Carlo Rota, a friend of Ulam:
“Riesz’s example is well worth following today.”
Related material: Misunderstanding in the Theory of Design and Geometry for Jews.
For a different approach to ethnicity and the number nine that is also “geometrical in nature,” see The Pope in Plato’s Cave and the four entries preceding it, as well as A Study in Art Education.
Comments Off on Sunday November 16, 2008
Wednesday, September 10, 2008
Wednesday September 10, 2008
Hitler on
Democracy
Democracy
The Fuehrer's wisdom seems especially appropriate today, in light of John McCain's recent "sex education for kindergarteners" and "lipstick" ads:
"… thus in the primitive simplicity of their minds they more readily fall victims to the big lie than the small lie, since they themselves often tell small lies in little matters but would be ashamed to resort to large-scale falsehoods. It would never come into their heads to fabricate colossal untruths, and they would not believe that others could have the impudence to distort the truth so infamously…. The grossly impudent lie always leaves traces behind it, even after it has been nailed down."
"I'm Joseph Goebbels, and
I approve this message."
Comments Off on Wednesday September 10, 2008
Saturday, August 16, 2008
Saturday August 16, 2008
Seeing the Finite Structure
The following supplies some context for remarks of Halmos on combinatorics.
From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–
Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–
“Part II: In Touch with God*“– on pp. 27-28:
The Root of All Deep Mathematics
“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.
Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.
Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.
Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”
Finite Structure
on a Book Cover:
on a Book Cover:
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990
Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.
Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.
* See the Log24 entries following Halmos’s death.
Comments Off on Saturday August 16, 2008
Tuesday, October 16, 2007
Tuesday October 16, 2007
In memory of
Harish-Chandra,
who died at 60
on this date in 1983 
Harish-Chandra in 1981
(Photo by Herman Landshof)
Harish-Chandra,
who died at 60
on this date in 1983
Harish-Chandra in 1981
(Photo by Herman Landshof)
Recent Log24 entries have parodied the use of the phrase “deep beauty” as the title of the Oct. 3-4 physics symposium of that name, which was supported by a grant from the John Templeton Foundation and sponsored by the Department of Philosophy at Princeton University.
Such parody was in part suggested by the symposium’s sources of financial and academic support. This support had, in the view of some, the effect of linking the symposium’s topic, the mathematics of quantum theory, with both religion (the Templeton Foundation) and philosophy (a field sometimes associated in popular thought– though not at Princeton— with quantum mysticism.)
As a corrective to the previous parodies here, the following material on the mathematician Harish-Chandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.
MacTutor History of Mathematics:
“Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.”
R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):
“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semi-simple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”
“It is difficult to communicate the grandeur of Harish-Chandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”
For deeper views of his work, see
- Rebecca A. Herb, “Harish-Chandra and His Work” (pdf), Bulletin of the American Mathematical Society, July 1991, and
- R. P. Langlands, “Harish-Chandra, 1923-1983” (pdf, 28 pp., Royal Society memoir, 1985)
Comments Off on Tuesday October 16, 2007
Saturday, March 10, 2007
Saturday March 10, 2007
The Logic of Dreams
From A Beautiful Mind–
“How could you,” began Mackey, “how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?”
Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. “Because,” Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, “the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.”
Ideas:
These numbers may, in the mad way so well portrayed by Sylvia Nasar in the above book, be regarded as telling a story… a story that should, of course, not be taken too seriously.
Friday’s New York numbers (midday 214, evening 711) suggest the dates 2/14 and 7/11. Clicking on these dates will lead the reader to Log24 entries featuring, among others, T. S. Eliot and Stephen King– two authors not unacquainted with the bizarre logic of dreams.
A link in the 7/11 entry leads to a remark of Noel Gray on Plato’s Meno and “graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave.”
Also Friday: an example of graphic austerity– indeed, Gray graphic austerity– in Log24:
This illustration refers to chess rather than to geometry, and to the mind of an addict rather than to that of a slave, but chess and geometry, like addiction and slavery, are not unrelated.
Friday’s Pennsylvania numbers, midday 429 and evening 038, suggest that the story includes, appropriately enough in view of the above Beautiful Mind excerpt, Mackey himself. The midday number suggests the date 4/29, which at Log24 leads to an entry in memory of Mackey.
(Related material: the Harvard Gazette of April 6, 2006, “Mathematician George W. Mackey, 90: Obituary“– “A memorial service will be held at Harvard’s Memorial Church on April 29 at 2 p.m.“)
Friday’s Pennsylvania evening number 038 tells two other parts of the story involving Mackey…
As Mackey himself might hope, the number may be regarded as a reference to the 38 impressive pages of Varadarajan’s “Mackey Memorial Lecture” (pdf).
More in the spirit of Nash, 38 may also be taken as a reference to Harvard’s old postal address, Cambridge 38, and to the year, 1938, that Mackey entered graduate study at Harvard, having completed his undergraduate studies at what is now Rice University.
Returning to the concept of graphic austerity, we may further simplify the already abstract chessboard figure above to obtain an illustration that has been called both “the field of reason” and “the Garden of Apollo” by an architect, John Outram, discussing his work at Mackey’s undergraduate alma mater:
Let us hope that Mackey,
a devotee of reason,
is now enjoying the company
of Apollo rather than that of
Tom O’Bedlam:
For John Nash on his birthday:
I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.
Comments Off on Saturday March 10, 2007
Saturday, December 16, 2006
Saturday December 16, 2006
Cubism1 as Multispeech2
— From Pedagogy, Praxis, Ulysses
A quotation omitted from the above excerpt:
In Ulysses, there is "… the same quality of simultaneity as in cubist collage. Thus, for example, Bloom surveys the tombstones at Paddy Dignam's funeral and, in the midst of platitudinous and humorous thoughts, remembers Molly 'wanting to do it at the window'…."
Related material from quotations at the poetry journal eratio:
"The guiding law of the great variations in painting is one of disturbing simplicity. First things are painted; then, sensations; finally, ideas. This means that in the beginning the artist's attention was fixed on external reality; then, on the subjective; finally, on the intrasubjective. These three stages are three points on a straight line."
— Jose Ortega y Gasset ("On Point of View in the Arts," an essay on the development of cubism)
Related material on
tombstones and windows:
Geometry's Tombstones,
Galois's Window, and
Architecture of Eternity.
See also the following part
of the eratio quotations:
Quotations arranged by
Gregory Vincent St. Thomasino
Sunday, February 12, 2006
Sunday February 12, 2006
Proposition
“… a new nation, conceived in liberty and dedicated to the proposition that all men are created equal”
— Speech, A. Lincoln, Nov. 19, 1863
Some are less equal than others.
Proof:
Jacques Herbrand, born on this date in 1908.
“Herbrand… worked on field theory, considering abelian extensions of algebraic number fields. In the few months on which he worked on this topic, Herbrand published ten papers. These papers simplify proofs of results by Kronecker, Heinrich Weber, Hilbert, Takagi and Artin. Herbrand also generalised some of the results by these workers in class field theory as well as proving some important new theorems of his own.” –MacTutor
See
Wednesday, February 1, 2006
Wednesday February 1, 2006
“Good Night and Good Luck”
— Morgan Freeman, closing remarks
at the Screen Actors Guild Awards
on January 29, 2006
 A History of Death |
 |
Review by Stephen Hunter
of “Good Night and Good Luck”–
“The film, therefore, is like
a child’s view of these events,
untroubled by complexity,
hungry for myth and simplicity.”
“A larger poem for a larger audience….
A mythological form, a festival sphere,
A great bosom, beard and being,
alive with age.”
— Wallace Stevens, quoted in
Log24, January 29, 2006
Stephen Hunter on Senator McCarthy:
He “forever tarnished
by association the reputations
of the security services
charged with keeping us safe
from the actual–
yes, Virginia, there was
such a thing– Red menace.”
Comments Off on Wednesday February 1, 2006
Tuesday, November 1, 2005
Tuesday November 1, 2005
The above seal is from an ad (pdf) for an Oct. 21 lecture, "The Nature of Space," by Sir Michael Atiyah, sponsored by the American Mathematical Society.
The picture in the seal is of Plato's Academy.
"The great philosopher Plato excluded from his Academy anyone who had not studied geometry. He would have been delighted to admit Sir Michael Atiyah, who was for a time Savilian Professor of Geometry at Oxford…"
Would he?
Sir Michael Atiyah's
Anti-Platonism
"Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance….
I tend to think that science and mathematics are ways the human mind looks and experiences– you cannot divorce the human mind from it. Mathematics is part of the human mind. The question whether there is a reality independent of the human mind, has no meaning– at least, we cannot answer it."
— Sir Michael Atiyah, interview in Oslo, May 2004
"For Plato, the Forms represent truth, or reality…. these Forms are independent of the mind: they are eternal, unchanging and perfect."
— Roy Jackson (pdf)
Atiyah's denial of a reality independent of the human mind may have something to do with religion:
"Socrates and Plato were considered 'Christians before Christ'; they paved the way for the coming of Christianity by providing it with philosophical and theoretical foundations that would be acceptable to the western mind.
In the analogy of the cave, the sun represents the Form of the Good. In the same way that the sun is the source of all things and gives light to them, the Form of the Good is over and above the other Forms, giving them light and allowing us to perceive them. Therefore, when you have awareness of the Form of the Good you have achieved true enlightenment. In Christianity, the Form of the Good becomes God: the source of all things."
— Roy Jackson, The God of Philosophy (pdf)
See also the previous entry.
Comments Off on Tuesday November 1, 2005
Tuesday, August 9, 2005
Tuesday August 9, 2005
Kaleidoscope
A new web page simplifies the Diamond 16 Puzzle and relates the resulting “kaleidoscope” to Hesse’s Bead Game.
Comments Off on Tuesday August 9, 2005
Tuesday, June 21, 2005
Tuesday June 21, 2005
Art History
“I studied with Reinhardt and I found that a fantastic course. I think he was really very stimulating….
Art history was very personal through the eyes of Ad Reinhardt.”
— Robert Morris,
Smithsonian Archives of American Art
Related material:
“The Road to Simplicity Followed by Merton’s Friends: Ad Reinhardt and Robert Lax” in The Merton Annual 13 (2000) 245-256, by Paul J. Spaeth, library director at St. Bonaventure University
The Merton here is Trappist monk Thomas Merton. Here is Merton in a letter to poet Robert Lax on the death of their friend Ad Reinhardt, sometimes called the “black monk” of abstract art:
“Make Mass beautiful silence like big black picture speaking requiem. Tears in the shadows of hermit hatch requiems blue black tone. Sorrows for Ad in the oblation quiet peace request rest. Tomorrow is solemns in the hermit hatch for old lutheran reinhardt commie paintblack… Tomorrow is the eternal solemns and the barefoots and the ashes and the masses, oldstyle liturgy masses without the colonels… Just old black quiet requiems in hermit hatch with decent sorrows good by college chum.”
— from J. S. Porter, “Farewell to a Monk,”
Antigonish Review, Winter 1997
Comments Off on Tuesday June 21, 2005
Thursday, January 27, 2005
Thursday January 27, 2005
Crystal Night
Mies van der Rohe:
Mies in Berlin
Winner of
The Society of Architectural Historians
2002 Philip Johnson Award
for Excellence
Exhibition Catalog
"Published to accompany
a groundbreaking 2001 exhibition at
The Museum of Modern Art, New York."
a groundbreaking 2001 exhibition at
The Museum of Modern Art, New York."
"It would have been wiser for the new MoMA catalog… to have addressed the issue of his politics…. By ignoring such a central subject… the show gives off a mild stench of cover-up…. Only the German-born Rosemarie Haag Bletter (full disclosure: she is my wife) alludes to the verboten topic in her [catalog] essay on Mies's flirtation with crystal imagery, drawing a sharp parallel between the architect's extensive use of Kristallglas (plate glass) and the ensuing devastation of Kristallnacht, which erupted just three months after he left for the States."
Also from Filler's essay:
"Mies's rigorously simplified structures, typified by grids of steel and glass and an absence of applied ornament, represented the Platonic ideal of modernism for many people."
For more on aesthetics, see the
Log24.net entry of Tuesday noon,
Log24.net entry of Tuesday noon,
For more on a Platonic ideal of sorts,
see the following figure in two versions:
see the following figure in two versions:
Version A, from Plato's Meno and
Diamond Theory,
Diamond Theory,
and Version B,
from the date of Johnson's death
at his "famous crystalline box."
Was less more?
Comments Off on Thursday January 27, 2005
Friday, September 17, 2004
Friday September 17, 2004
God is in…
The Details
From an entry for Aug. 19, 2003 on
conciseness, simplicity, and objectivity:
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest. Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.
For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at |
On Harvard and psychiatry: see
The Crimson Passion:
A Drama at Mardi Gras
(February 24, 2004)
This is a reductio ad absurdum of the Harvard philosophy so eloquently described by Alston Chase in his study of Harvard and the making of the Unabomber, Ted Kaczynski. Kaczynski's time at Harvard overlapped slightly with mine, so I may have seen him in Cambridge at some point. Chase writes that at Harvard, the Unabomber "absorbed the message of positivism, which demanded value-neutral reasoning and preached that (as Kaczynski would later express it in his journal) 'there is no logical justification for morality.'" I was less impressed by Harvard positivism, although I did benefit from a course in symbolic logic from Quine. At that time– the early 60's– little remained at Harvard of what Robert Stone has called "our secret culture," that of the founding Puritans– exemplified by Cotton and Increase Mather.
From Robert Stone, A Flag for Sunrise:
"Our secret culture is as frivolous as a willow on a tombstone. It's a wonderful thing– or it was. It was strong and dreadful, it was majestic and ruthless. It was a stranger to pity. And it's not for sale, ladies and gentlemen."
Some traces of that culture:
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A web page
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A contemporary
Click on pictures for details. |
A more appealing view of faith was offered by PBS on Wednesday night, the beginning of this year's High Holy Days:
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Armand Nicholi: But how can you believe something that you don't think is true, I mean, certainly, an intelligent person can't embrace something that they don't think is true — that there's something about us that would object to that.
Jeremy Fraiberg: Well, the answer is, they probably do believe it's true. Armand Nicholi: But how do they get there? See, that's why both Freud and Lewis was very interested in that one basic question. Is there an intelligence beyond the universe? And how do we answer that question? And how do we arrive at the answer of that question? Michael Shermer: Well, in a way this is an empirical question, right? Either there is or there isn't. Armand Nicholi: Exactly. Michael Shermer: And either we can figure it out or we can't, and therefore, you just take the leap of faith or you don't. Armand Nicholi: Yeah, now how can we figure it out? Winifred Gallagher: I think something that was perhaps not as common in their day as is common now — this idea that we're acting as if belief and unbelief were two really radically black and white different things, and I think for most people, there's a very — it's a very fuzzy line, so that — Margaret Klenck: It's always a struggle. Winifred Gallagher: Rather than — I think there's some days I believe, and some days I don't believe so much, or maybe some days I don't believe at all. Doug Holladay: Some hours. Winifred Gallagher: It's a, it's a process. And I think for me the big developmental step in my spiritual life was that — in some way that I can't understand or explain that God is right here right now all the time, everywhere. Armand Nicholi: How do you experience that? Winifred Gallagher: I experience it through a glass darkly, I experience it in little bursts. I think my understanding of it is that it's, it's always true, and sometimes I can see it and sometimes I can't. Or sometimes I remember that it's true, and then everything is in Technicolor. And then most of the time it's not, and I have to go on faith until the next time I can perhaps see it again. I think of a divine reality, an ultimate reality, uh, would be my definition of God. |
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Gallagher seemed to be the only participant in the PBS discussion that came close to the Montessori ideals of conciseness, simplicity, and objectivity. Dr. Montessori intended these as ideals for teachers, but they seem also to be excellent religious values. Just as the willow-tombstone seems suited to Geoffrey Hill's style, the Pythagorean sangaku pictured above seems appropriate to the admirable Gallagher.