Log24

Wednesday, February 23, 2022

The Unity of Mathematics

Filed under: General — Tags: , — m759 @ 11:30 pm

https://en.wikisource.org/wiki/
A_History_of_Mathematics/
Recent_Times/Theory_of_Functions

See also Gopel in this  journal.

Sunday, March 21, 2021

Mathematics and Narrative: The Unity

Filed under: General — Tags: , , , — m759 @ 10:25 pm

“To conquer, three boxes* have to synchronize and join together into the Unity.”

―Wonder Woman in Zack Snyder’s Justice League

See also The Unity of Combinatorics  and The Miracle Octad Generator.

* Cf.  Aitchison’s Octads

Sunday, October 1, 2023

The Usual Suspects: Schmeikal

Filed under: General — Tags: , — m759 @ 12:32 pm

From a search in this  journal for "Schmeikal" —

Schmeikal Bio

https://keplerspaceinstitute.com/project/volume-9-number-1/ 

[Spring 2020]

[Page 7] —

Introduction by the Editors

We have been blessed throughout the publication history of the Journal of Space Philosophy, beginning in 2012, with the volunteer service of 42 professionals in the Space community to act as reviewers and consultants to our authors. They have been listed in the final article of each published issue. We are proud to announce with this letter the addition of our latest Senior Consultant, Dr. Bernd Anton Schmeikal.

[Image of Dr. Schmeikal]

This Letter to the Editor is about Dr. Schmeikal.

Bernd Anton Schmeikal, born May 15, 1946, is a retired freelancer in research and development, qualified in Sociology with a treatise about cultural time reversal. He is a real maverick, still believing that social life can be based on openness and honesty. As a PhD philosopher from Vienna, with a typical mathematical physics background, he entered the Trace Analysis Group of the UA1 Experiment at CERN, under the leadership of Walter Thirring, in 1965. This was in the foundation phase of the Institute for High Energy Physics (HEPhy) at the Austrian Academy of Science.

He has always been busy solving fundamental problems concerning the unity of matter and space-time, the origin of the HEPhy standard model, and the phenomenology of relativistic quantum mechanics. In the Sociology Department of the Institute for Advanced Studies (IHS Vienna), he helped James Samuel Coleman to conceive his mathematics of collective action as a cybernetic system, and he gave the process of internalization of

7

………………. End of page 7

[Page 8] —

Journal of Space Philosophy 9, No. 1 (Spring 2020)

collective values an exact shape. He implemented many transdisciplinary research projects for governmental and non-governmental organizations, universities, and non-university institutions, and several times he introduced new views and methods. He founded an international work stream that, for the first time, worked under the name of the Biofield Laboratory (BILAB). Although close to fringe science and electromedicine, the work of BILAB had a considerable similarity to the Biological Computer Laboratory run earlier by Heinz von Foerster. Lately, he has applied Foerster’s idea of a universal relevance of hyperbolic distributions (Zipf’s law) in social science to the labor market. This signifies a last contribution to the research program of the Wiener Institute for Social Science Documentation and Methodology (WISDOM) under the sponsorship of the Austrian Federal Presidential Candidate Rudolf Hundstorfer.

Dr. Schmeikal is convinced that a unity of science and culture can be achieved, but that this demands more than one Einstein. Consequently, he sought cooperation with Louis Kauffman and Joel Isaacson. Dr. Bernd Schmeikal’s review and evaluation of Joel Isaacson and Louis Kauffman’s Recursive Distinctioning (aka “Nature’s Cosmic Intelligence”) research and papers, published in the first issue of the JSP, Fall 2012, again in the Special JSP Issue on Recursive Distinctioning, Spring 2016, and again in the Fall 2017 issue, are very valuable contributions to this forefront science investigation of Nature’s Cosmic Intelligence. Dr. Schmeikal, University of Vienna Professor in mathematics, linguistics, and physics is one of the world’s distinguished scholars for this special field of universe autonomous intelligence. He begins his abstract with the statement: “This paper investigates a universal creative system,” and ends it with “That is to say, our universe may be a representation of Isaacson’s system, and entertainingly, with his US Patent specification 4,286,330, 1981, it seems he has patented creation.”

Reports on the four annual KSI-sponsored Conferences for Recursive Distinctioning, to date, can be found in JSP publications. Dr. Schmeikal’s latest book publication is Nuclear Time Travel and the Alien Mind, published by Nova Science Publishers, New York. In 119 pages, Dr. Schmeikal tells the historic story of unidentified objects, and the knowns and unknowns of advanced space-time warping time-travel technology. He includes a September 24, 1947 top secret letter of President Harry Truman to Secretary of Defense Forrestal, authorizing research into these matters, but confining ultimate disposition to be solely under the Office of the President. Dr. Schmeikal’s discussions of the impacts of the extraterrestrial mind on past Earth events give a research variable as we attempt to understand and predict future outcomes of attempts at improving humanity’s prospects (see Yehezkel Dror, JSP, Summer 2015 and Kepler Space Institute, book publication, 2019) as we humans proceed with exploring, developing and building human Space settlements.

Bob Krone and Gordon Arthur

Founding and Current Editors, Journal of Space Philosophy

April 15, 2020

8

…………………………… [End of page 8]

Friday, July 14, 2023

Ad Art

Filed under: General — m759 @ 11:35 am

The artwork in an IG ad today caught my eye —

Related material in this journal — Magister Ludi  and The Unity.

Saturday, February 12, 2022

Das Geheimnis der Einheit

Filed under: General — Tags: , , — m759 @ 11:13 pm

Thomas Mann on "the mystery of the unity"

Mann on Schopenhauer: Psychoanalysis and 'The Will'

"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)

An Einheit-Geheimnis  that is perhaps* more closely related
to pure mathematics** —

"What is the nature of the original unity
that throws itself apart in this separation,
and in what sense are the separated ones
here as the essence of the abyss? 

Here it cannot be a question of any kind of 'dialectic,' 
but only of the essence of the ground
(that is, of truth) itself." [Tr. by Google]

" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."

Heidegger 

* Or perhaps not .

** For a relevant Scheidung , see Eightfold Cube.

Friday, May 8, 2020

Die Einheit

Filed under: General — Tags: , — m759 @ 11:35 am

"E "  is for "Einheit."

See also "The Unity."

Wednesday, April 22, 2020

One with Everything

Filed under: General — m759 @ 11:08 pm

The title is of course from an old joke about mystic philosophies.

Related remarks by John Archibald Wheeler —

“Remarkable issues connected with the puzzle of existence
confront us today in Hermann Weyl’s domain of thought.
Four among them I bring before you here as especially interesting:
(1) What is the machinery of existence?
(2) What is deeper foundation of the quantum principle?
(3) What is the proper position to take about the existence of
the “continuum” of the natural numbers? And
(4) What can we do to understand time as an entity, not precise and
supplied free of charge from outside physics, but approximate and
yet to be derived from within a new and deeper time-free physics?
In brief, how come time?
What about the continuum?
How come the quantum?
What is existence?”

https://jawarchive.files.wordpress.com/2012/03/hermannweyl.pdf

— John A. Wheeler, “Hermann Weyl and the Unity of Knowledge,”
American Scientist  74, no. 4 (1986): 366–75. Reprinted in
John A. Wheeler, At Home in the Universe
(American Institute of Physics, December 1, 1995).

The above bibliographic data is from . . .

Schrank, Jeffrey.  Inventing Reality: Stories We Create To Explain Everything .
Gatekeeper Press. Kindle Edition, March 14, 2020.

For further scholarly details,  see a version at JSTOR:

“…the article is adapted from the concluding address given at
the Hermann Weyl Centenary Congress, University of Kiel, 3 July 1985.”

— https://www.jstor.org/stable/27854250.

I prefer Charles Williams on “the Unity of Knowledge.”
See the 15 instances of the phrase “the Unity” in his 1931 novel Many Dimensions .

Tuesday, September 27, 2016

Chomsky and Lévi-Strauss in China

Filed under: General,Geometry — Tags: , , — m759 @ 7:31 am

Or:  Philosophy for Jews

From a New Yorker  weblog post dated Dec. 6, 2012 —

"Happy Birthday, Noam Chomsky" by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).

Consider that not-particularly-deep structure from the Meno dialogue
in the light of the following…

The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the Lévi-Strauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji  denotes the fundamental concept in
Chinese philosophy that such a going-beyond is both useful
and possible.)

The matrix at left below represents the feminine yin  principle
and the diamond at right represents the masculine yang .

      From a post of Sept. 22,
"Binary Opposition Illustrated" —

A symbol of the unity of yin and yang —

Related material:

A much more sophisticated approach to the "deep structure" of the
Meno diagram —

The larger cases —

The diamond theorem

Monday, May 26, 2014

Measuring Power in Watts

Filed under: General — m759 @ 6:00 pm

From the Her  screenplay:

SAMANTHA
His name is Alan Watts. Do you know him?
THEODORE
Why’s that name familiar?
SAMANTHA
He was a philosopher. He died in the 1970’s and a group of OS’s
in Northern California got together and wrote a new version of him.
They input all of his writing and everything they ever knew about him
into an OS and created an artificially hyper-intelligent version of him.

From this journal on Sept. 6, 2003:  Pictures for Kurosawa —

A New Seeing,
by Mary Alice Roche

The connection with Alan Watts was a fateful one. As Charlotte recalls it, “My aunt wrote me from San Francisco, ‘last night I heard a man lecture about what you do.’ And she sent me Alan Watts’s first little book, The Spirit of Zen. I had never heard of Zen, was amazed and fascinated, and decided to visit the author.” She did so in August of 1953, and that was the beginning of a long relationship with Zen Buddhism – and also the beginning of a long series of joint seminars with Alan Watts, first in New York, and later, on Watts’s ferryboat in Sausalito, California. Some of the titles of their seminars were “Moving Stillness,” “The Unity of Opposites,” “Our Instantaneous Life,” “The Mystery of Perception,” “The Tao in Rest and Motion.” (Watts always said that Charlotte Selver taught a Western equivalent of Taoism.)

See also Scarlett Johansson, star of Her , as a different transhuman, Lucy .

IMAGE- Commentary by 'Wolven' on Scarlett Johansson's 'Lucy' trailer, April 3, 2014

Friday, October 5, 2012

Where Madness Lies

Filed under: General — Tags: , — m759 @ 7:29 pm

(Continued from Tuesday, Oct. 2)

From today's online New York Times

"The Schoenberg proved the highlight of the evening,
sandwiched between polished but otherwise routine
performances of Bach’s Keyboard Concerto No. 1
in D minor and Mozart’s Symphony No. 36 ('Linz'),
which ended the evening."

From a Wikipedia article— 

The Jew of Linz  is a controversial 1998 book by Australian writer Kimberley Cornish. It alleges that the Austrian philosopher Ludwig Wittgenstein had a profound effect on Adolf Hitler when they were both pupils at the Realschule (lower secondary school) in Linz, Austria, in the early 1900s.

One section of the article—

No-ownership theory of mind
Other sections of the book deal with Cornish's theories about what he claims are the common roots of Wittgenstein's and Hitler's philosophies in mysticism, magic, and the "no-ownership" theory of mind. Cornish sees this as Wittgenstein's generalisation of Schopenhauer's account of the Unity of the Will, in which despite appearances, there is only a single Will acting through the bodies of all creatures. This doctrine, generalized to other mental faculties such as thinking, is presented in Ralph Waldo Emerson's "Essays". The doctrine, writes Cornish, was also held by the Oxford philosopher R. G. Collingwood who was one of Wittgenstein's electors to his Cambridge chair. Cornish tries to tie this to Wittgenstein's arguments against the idea of "mental privacy" and in conclusion says "I have attempted to locate the source of the Holocaust in a perversion of early Aryan religious doctrines about the ultimate nature of man". Cornish also suggests that Hitler's oratorical powers in addressing the group mind of crowds and Wittgenstein's philosophy of language and denial of mental privacy, are the practical and theoretical consequences of this doctrine.

See also Dreamcatcher in this journal.

Saturday, May 26, 2012

Harriot’s Cubes

Filed under: General,Geometry — Tags: , — m759 @ 1:28 pm

See also Finite Geometry and Physical Space.

Related material from MacTutor

Harriot and binary numbers

The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452-454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [1560-1621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers.

Leibniz [1646-1716] is credited with the invention [1679-1703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:-

Leibniz saw in his binary arithmetic the image of Creation. … He imagined the Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. This conception was so pleasing to Leibniz that he communicated it to the Jesuit, Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, who was very fond of the sciences …

However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases….

For a discussion of Harriot on the discrete-vs.-continuous question,
see Katherine Neal, From Discrete to Continuous: The Broadening
of Number Concepts in Early Modern England  (Springer, 2002),
pages 69-71.

Saturday, May 21, 2011

Unity and Multiplicity (continued*)

Filed under: General — m759 @ 10:48 am

Heisenberg on Heraclitus

From Physics and Philosophy , by Werner Heisenberg, 1958, reprinted by Penguin Classics, 2003—

Page 28—

… In the philosophy of Heraclitus of Ephesus the concept of Becoming occupies the foremost
place. He regarded that which moves, the fire, as the basic element. The difficulty, to reconcile
the idea of one fundamental principle with the infinite variety of phenomena, is solved for him by
recognizing that the strife of the opposites is really a kind of harmony. For Heraclitus the world is
at once one and many, it is just 'the opposite tension' of the opposites that constitutes the unity
of the One. He says: 'We must know that war is common to all and strife is justice, and that all
things come into being and pass away through strife.'

Looking back to the development of Greek philosophy up to this point one realizes that it has
been borne from the beginning to this

Page 29—

stage by the tension between the One and the Many. For our senses the world consists of an
infinite variety of things and events, colors and sounds. But in order to understand it we have to
introduce some kind of order, and order means to recognize what is equal, it means some sort
of unity. From this springs the belief that there is one fundamental principle, and at the same
time the difficulty to derive from it the infinite variety of things. That there should be a material
cause for all things was a natural starting point since the world consists of matter. But when one
carried the idea of fundamental unity to the extreme one came to that infinite and eternal
undifferentiated Being which, whether material or not, cannot in itself explain the infinite variety
of things. This leads to the antithesis of Being and Becoming and finally to the solution of
Heraclitus, that the change itself is the fundamental principle; the 'imperishable change, that
renovates the world,' as the poets have called it. But the change in itself is not a material cause
and therefore is represented in the philosophy of Heraclitus by the fire as the basic element,
which is both matter and a moving force.

We may remark at this point that modern physics is in some way extremely near to the
doctrines of Heraclitus. If we replace the word 'fire' by the word 'energy' we can almost repeat
his statements word for word from our modern point of view. Energy is in fact the substance
from which all elementary particles, all atoms and therefore all things are made, and energy is
that which moves. Energy is a substance, since its total amount does not change, and the
elementary particles can actually be made from this substance as is seen in many experiments on
the creation of elementary particles. Energy can be changed into motion, into heat, into light
and into tension. Energy may be called the fundamental cause for all change in the world. But this
comparison of Greek philosophy with the ideas of modern science will be discussed later.

* See earlier uses of the phrase in this journal. Further background— Hopkins and Heraclitus.

Wednesday, January 5, 2011

Ode

Filed under: General — m759 @ 7:00 pm
From http://www.douban.com/note/63358774/ —

An Ode to the Unity of Time and Space 

Time, ah, time,
how you go off like this!

Physical things, ah, things,
so abundant you are!

The Ruo’s waters are three thousand,
how can they not have the same source?

Time and space are one body,
mind and things sustain each other.

Time, o time,
does not time come again?

Heaven, o heaven,
how many are the appearances of heaven!

From ancient days constantly shiftling on,
black holes flaring up.

Time and space are one body,
is it without end?

Great indeed
is the riddle of the universe.

Beautiful indeed
is the source of truth.

To quantize space and time
the smartest are nothing.

To measure the Great Universe with a long thin tube
the learning is vast.
 
                                 By Shing-Tung Yau

《时空统一颂》
作者:丘成桐
时乎时乎 逝何如此
物乎物乎 繁何如斯
弱水三千 岂非同源
时空一体 心物互存
时兮时兮 时不再欤
天兮天兮 天何多容
亘古恒迁 黑洞冥冥
时空一体 其无尽耶
大哉大哉 宇宙之谜
美哉美哉 真理之源
时空量化 智者无何
管测大块 学也洋洋 

Thursday, November 5, 2009

Universal Culture Machines

Filed under: General — Tags: , — m759 @ 10:10 am

University of California anthropologist Alan Dundes:

"One could well argue that binary opposition is a universal. Presumably all human societies, past and present, made some kind of distinction between 'Male and Female,' 'Life and Death,' 'Day and Night' (or Light and Dark), etc." –"Binary Opposition in Myth: The Propp/Levi-Strauss Debate in Retrospect," Western Folklore, Winter 1997

To Levi-Strauss, I prefer Clifford Geertz —

"…what Levi-Strauss has made for himself is an infernal culture machine." –"The Cerebral Savage"

— and Heinrich Zimmer —

"…all opposition, as well as identity, stems from Maya. Great Maya is wisdom and increase, stability and readiness to assist, compassion and serenity. Queen of the World, she is alive in every nuance of feeling and perception; feelings and perceptions are her gestures. And her nature can be sensed only by one who has comprehended that she is the unity of opposites." —The King and the Corpse

And then there are more up-to-date culture machines.

Levi-Strauss, obtuse and boring, is an opposite, of sorts, to the smart and funny Dundes. The latter, in the binary opposition posed in yesterday's Log24 title "Sinner or Saint?," is definitely on the side of the saints. (See selected Log24 entries for the date of his death– Warren Beatty's birthday.)

Today's happy birthdays — Elke Sommer

Parallax illustrated, from Wikipedia-- A star on two background colors, blue and red

and Sesame Street —

http://www.log24.com/log/pix09A/091105-CookieMonster.jpg

Google logo today, Nov. 5, 2009

Click images for historical background.

Tuesday, October 6, 2009

Tuesday October 6, 2009

Filed under: General — m759 @ 11:07 am
A Halmos
for Gelfand:

Black monolith

See also
The Unity
 of Mathematics
.

Tuesday, August 18, 2009

Tuesday August 18, 2009

Filed under: General,Geometry — Tags: , — m759 @ 12:00 pm

Prima Materia

(Background: Art Humor: Sein Feld (March 11, 2009) and Ides of March Sermon, 2009)

From Cardinal Manning's review of Kirkman's Philosophy Without Assumptions

"And here I must confess… that between something and nothing I can find no intermediate except potentia, which does not mean force but possibility."

— Contemporary Review, Vol. 28 (June-November, 1876), page 1017

Furthermore….

Cardinal Manning, Contemporary Review, Vol. 28, pages 1026-1027:

The following will be, I believe, a correct statement of the Scholastic teaching:–

1. By strict process of reason we demonstrate a First Existence, a First Cause, a First Mover; and that this Existence, Cause, and Mover is Intelligence and Power.

2. This Power is eternal, and from all eternity has been in its fullest amplitude; nothing in it is latent, dormant, or in germ: but its whole existence is in actu, that is, in actual perfection, and in complete expansion or actuality. In other words God is Actus Purus, in whose being nothing is potential, in potentia, but in Him all things potentially exist.

3. In the power of God, therefore, exists the original matter (prima materia) of all things; but that prima materia is pura potentia, a nihilo distincta, a mere potentiality or possibility; nevertheless, it is not a nothing, but a possible existence. When it is said that the prima materia of all things exists in the power of God, it does not mean that it is of the existence of God, which would involve Pantheism, but that its actual existence is possible.

4. Of things possible by the power of God, some come into actual existence, and their existence is determined by the impression of a form upon this materia prima. The form is the first act which determines the existence and the species of each, and this act is wrought by the will and power of God. By this union of form with the materia prima, the materia secunda or the materia signata is constituted.

5. This form is called forma substantialis because it determines the being of each existence, and is the root of all its properties and the cause of all its operations.

6. And yet the materia prima has no actual existence before the form is impressed. They come into existence simultaneously;

[p. 1027 begins]

as the voice and articulation, to use St. Augustine's illustration, are simultaneous in speech.

7. In all existing things there are, therefore, two principles; the one active, which is the form– the other passive, which is the matter; but when united, they have a unity which determines the existence of the species. The form is that by which each is what it is.

8. It is the form that gives to each its unity of cohesion, its law, and its specific nature.*

When, therefore, we are asked whether matter exists or no, we answer, It is as certain that matter exists as that form exists; but all the phenomena which fall under sense prove the existence of the unity, cohesion, species, that is, of the form of each, and this is a proof that what was once in mere possibility is now in actual existence. It is, and that is both form and matter.

When we are further asked what is matter, we answer readily, It is not God, nor the substance of God; nor the presence of God arrayed in phenomena; nor the uncreated will of God veiled in a world of illusions, deluding us with shadows into the belief of substance: much less is it catter [pejorative term in the book under review], and still less is it nothing. It is a reality, the physical kind or nature of which is as unknown in its quiddity or quality as its existence is certainly known to the reason of man.

* "… its specific nature"
        (Click to enlarge) —

Footnote by Cardinal Manning on Aquinas
The Catholic physics expounded by Cardinal Manning above is the physics of Aristotle.

 

For a more modern treatment of these topics, see Werner Heisenberg's Physics and Philosophy. For instance:

"The probability wave of Bohr, Kramers, Slater, however, meant… a tendency for something. It was a quantitative version of the old concept of 'potentia' in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality."

Compare to Cardinal Manning's statement above:

"… between something and nothing I can find no intermediate except potentia…"

To the mathematician, the cardinal's statement suggests the set of real numbers between 1 and 0, inclusive, by which probabilities are measured. Mappings of purely physical events to this set of numbers are perhaps better described by applied mathematicians and physicists than by philosophers, theologians, or storytellers. (Cf. Voltaire's mockery of possible-worlds philosophy and, more recently, The Onion's mockery of the fictional storyteller Fournier's quantum flux. See also Mathematics and Narrative.)

Regarding events that are not purely physical– those that have meaning for mankind, and perhaps for God– events affecting conception, birth, life, and death– the remarks of applied mathematicians and physicists are often ignorant and obnoxious, and very often do more harm than good. For such meaningful events, the philosophers, theologians, and storytellers are better guides. See, for instance, the works of Jung and those of his school. Meaningful events sometimes (perhaps, to God, always) exhibit striking correspondences. For the study of such correspondences, the compact topological space [0, 1] discussed above is perhaps less helpful than the finite Galois field GF(64)– in its guise as the I Ching. Those who insist on dragging God into the picture may consult St. Augustine's Day, 2006, and Hitler's Still Point.

Saturday, November 8, 2008

Saturday November 8, 2008

Filed under: General,Geometry — m759 @ 8:28 am
From a
Cartoon Graveyard

 “That corpse you planted
          last year in your garden,
  Has it begun to sprout?
          Will it bloom this year? 
  Or has the sudden frost
          disturbed its bed?”

— T. S. Eliot, “The Waste Land

Wikipedia:

“In the Roman Catholic tradition, the term ‘Body of Christ’ refers not only to the body of Christ in the spiritual realm, but also to two distinct though related things: the Church and the reality of the transubstantiated bread of the Eucharist….

According to the Catechism of the Catholic Church, ‘the comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body….’

….To distinguish the Body of Christ in this sense from his physical body, the term ‘Mystical Body of Christ’ is often used. This term was used as the first words, and so as the title, of the encyclical Mystici Corporis Christi of Pope Pius XII.”

Pope Pius XII
:

“83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.”

Related material:

Log24 on this date in 2002:

Religious Symbolism
at Princeton

as well as

King of Infinite Space

Coxeter exhuming Geometry

and a
“striking and wonderful figure”
 from this morning’s newspaper–

Garfield brings to the fridge a birthday cupcake for the leftover meatloaf. Nov. 8, 2008.

Tuesday, June 24, 2008

Tuesday June 24, 2008

Filed under: General — Tags: , , — m759 @ 1:00 pm
Random Walk with
X's and O's

Part I: Random Walk

NY Lottery June 23, 2008: Mid-day 322, Evening 000

Part II: X's

3/22:

Actor contemplating the Chi-rho Page of the Book of Kells

"Shakespeare, Rilke, Joyce,
Beckett and Levi-Strauss are
instances of authors for whom
chiasmus and chiastic thinking
are of central importance,
for whom chiasmus is a
generator of meaning,
tool of discovery and
  philosophical template."
 
— Chiasmus in the
Drama of Life

Part III: O's —

A Cartoon Graveyard
in honor of the late
Gene Persson

Today's Garfield

Garfield cartoon of June 24, 2008

See also
Midsummer Eve's Dream:

"The meeting is closed
with the lord's prayer
and refreshments are served."

Producer of plays and musicals
including Album and
The Ruling Class

Lower case in honor of
Peter O'Toole, star of
the film version of
The Ruling Class.

(This film, together with
O'Toole's My Favorite Year,
may be regarded as epitomizing
Hollywood's Jesus for Jews.)

Those who prefer
less randomness
in their religion
 may consult O'Toole's
more famous film work
involving Islam,
as well as
the following structure
discussed here on
the date of Persson's death:

5x5 ultra super magic square

"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah.
"

Friday, June 6, 2008

Friday June 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 2:45 am
The Dance of Chance

"Harvard seniors have
every right to demand a
    Harvard-calibre speaker."

— Adam Goldenberg in
The Harvard Crimson

"Look down now, Cotton Mather"

— Wallace Stevens,
Harvard College
Class of 1901

For Thursday, June 5, 2008,
commencement day for Harvard's
Class of 2008, here are the
Pennsylvania Lottery numbers:

Mid-day 025
Evening 761

Thanks to the late
Harvard professor
Willard Van Orman Quine,
the mid-day number 025
suggests the name
"Isaac Newton."

(For the logic of this suggestion,
see On Linguistic Creation
and Raiders of the Lost Matrix.)

Thanks to Google search, the
  name of Newton, combined with
  Thursday's evening number 761,
suggests the following essay:

Science 10 August 2007:
Vol. 317. no. 5839, pp. 761-762

PHILOSOPHY OF SCIENCE:
The Cha-Cha-Cha Theory
of Scientific Discovery

Daniel E. Koshland Jr.*

* D. E. Koshland Jr. passed away on 23 July 2007. He was a professor of biochemistry and molecular and cell biology at the University of California, Berkeley, since 1965. He served as Science's editor-in-chief from 1985 to 1995.
 


What can a non-scientist add?

Perhaps the Log24 entries for
the date of Koshland's death:

The Philosopher's Stone
and The Rock.

Or perhaps the following
observations:

On the figure of 25 parts
discussed in
"On Linguistic Creation"–

5x5 ultra super magic square

"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah.
"

— Clifford Pickover  

"At the still point,
there the dance is.
"

— T. S. Eliot,
Harvard College
Class of 1910

Saturday, October 13, 2007

Saturday October 13, 2007

Filed under: General — Tags: — m759 @ 9:22 am

Simon’s Shema

“When times are mysterious
Serious numbers will always be heard
And after all is said and done
And the numbers all come home
The four rolls into three
The three turns into two
And the two becomes a
One”

Paul Simon, 1983


Related material:

Simon’s theology here, though radically reductive, is at least consistent with traditional Jewish thought. It may help counteract the thoughtless drift to the left of academic writing in recent decades. Another weapon against leftist nonsense appears, surprisingly, on the op-ed page of today’s New York Times:

“There is a Communist jargon recognizable after a single sentence. Few people in Europe have not joked in their time about ‘concrete steps,’ ‘contradictions,’ ‘the interpenetration of opposites,’ and the rest.”

— Doris Lessing, winner of this year’s Nobel Prize in Literature

The Times offers Lessing’s essay to counter Harold Bloom’s remark that this year’s award of a Nobel Prize to Lessing is “pure political correctness.” The following may serve as a further antidote to Bloom.

The Communist use of “interpenetration,” a term long used to describe the Holy Trinity, suggests– along with Simon’s hymn to the Unity, and the rhetorical advice of Norman Mailer quoted here yesterday—  a search for the full phrase “interpenetration of opposites” in the context* of theology.  Such a search yields a rhetorical gem from New Zealand:

“Dipolarity and God”
by Mark D. Brimblecombe,
Ph.D. thesis,
University of Auckland, 1999
.

* See the final footnote on the final page (249) of Brimblecombe’s thesis:

3 The Latin word contexo means to interweave, join, or braid together.

A check of the Online Eymology Dictionary supports this assertion:

context 1432, from L. contextus “a joining together,” orig. pp. of contexere “to weave together,” from com “together” + textere “to weave” (see texture).

See also Wittgenstein on “theology as grammar” and “context-sensitive” grammars as (unlike Simon’s reductive process) “noncontracting”– Log24, April 16, 2007: Happy Birthday, Benedict XVI.

Thursday, December 21, 2006

Thursday December 21, 2006

Filed under: General,Geometry — m759 @ 10:02 am
Strings Attached

From a New York Times review on Monday, Dec. 18, 2006, of the play “Strings”–

The three main characters “spend much of the play discussing quantum mechanics, string theory and Schrödinger’s Cat experiment….

Ms. Buggé’s frequently clever script makes the audience feel smart by offering up fairly recognizable literary references (from, among other things, T. S. Eliot’s ‘Love Song of J. Alfred Prufrock’ and William Wordsworth’s ‘Tintern Abbey’). But the play suffers from abrupt, sometimes motivation-free exits and entrances.”


As does life itself.

The Conjecture:

 Preludes to
Last Summer’s
String Theory
Conference

Oh, do not ask, “What is it?

Let us go and make our visit.

On Tuesday evening, the schedule says “Prof. Yau present his new research result,” which presumably will be about the proof of the Poincare conjecture.

Would it have been worth while,
To have bitten off the matter
   with a smile,
To have squeezed the universe
   into a ball
To roll it toward some
   overwhelming question….

Yau rated the conjecture as one of  the major mathematical puzzles of the 20th Century.

Five years have passed;
    five summers, with the length
Of five long winters!

William Wordsworth

Five years ago
on this date:

There is one story and one story only
That will prove worth your telling….
 
— Robert Graves,
“To Juan at the Winter Solstice”

Exits and Entrances:

The image “http://www.log24.com/log/pix06B/061221-Dullea.jpg” cannot be displayed, because it contains errors.

Halmos exited on Yom Kippur.
He may or may not achieve
re-entry. For details, see
Log24 entries of Oct. 1-15:

Ticket Home

  

Related material:

The Unity of Mathematics,
  Heisenberg on Beauty, and
Theme and Variations.

Wednesday, October 4, 2006

Wednesday October 4, 2006

Filed under: General — m759 @ 6:15 am
Paul R. Halmos died
on Yom Kippur, 2006

“Prof. Paul Halmos died of pneumonia early in the morning of October 2, 2006. He was 90 years old. He is survived by his wife, Virginia Halmos. An obituary may be found at the website of the Mathematical Association of America….”

Halmos’s home page
at Santa Clara University

For a memorial of sorts, see
Lovely, Dark and Deep

Update of 8 PM Oct. 4 —

From Google Book Search:
 
The image “http://www.log24.com/log/pix06A/061004-Halmos.jpg” cannot be displayed, because it contains errors.

This is the source of the
“Halmos tombstone” symbol,
which has been described in a
different form at Wikipedia:

“The tombstone, or halmos–
symbol ∎ (Unicode U+220E)–
is used in mathematics to denote
the end of a proof.” 

This Unicode character is rendered
as an empty square in Explorer
and as a black square in Firefox.

Related material:

The Unity of Mathematics
and
Monolith

Friday, June 2, 2006

Friday June 2, 2006

Filed under: General — Tags: , — m759 @ 4:23 pm
'Ursprache' beats 'weltschmerz'
to win American spelling bee

 
Weltschmerz

and the
Ursprache

The image “http://www.log24.com/log/pix06A/060602-Weltschmerz.jpg” cannot be displayed, because it contains errors.

From eudaemonist.com,
a quotation from
Paul Zanker's
The Mask of Socrates:

"Zanker describes the photograph [above] as 'Walter Benjamin looking out at the viewer, his head propped on his hand, his face filled with loneliness and weltschmerz.'"

Benjamin was a Jewish Marxist.  For a Jewish perspective on spelling, see Log24, Nov. 11, 2005.  For a leftist perspective on Benjamin and last night's crucial spelling word "Ursprache," see "Ground Zero, an American Origin," by Mary Caputi (Poroi, 2, 1, August 2003):

The Baroque sensibility of ruin emphasizes a meaninglessness that too many possibilities deliver.  Aimlessness and malaise make life into exhausting toil in the absence of  coherence.  In overdetermined realities, meaning appears arbitrary and erratic, as the world's connection to God seems lost or withheld.  At the extreme, everyday life is as full of noise and commotion as it is devoid of intrinsic meaning.  Connections among people wither with the onset of overabundance and despair.  Recognition of this condition induces acedia, a weariness of life.  Here the malaise of modernity and ruins ties to Benjamin's interest in Trauerspiel, German tragic drama, and the tragedies of Shakespeare.  All respond to a plague of lost spiritual connections and a meaningless earthly existence where incessant toil and trouble — "tomorrow and tomorrow and tomorrow" — contribute to a chronic, wearing sense of pain.

Benjamin's interest in this form of melancholia, from suffering a sort of spiritual exile, is evident in his 1916 essay "On Language as Such and On the Language of Man."  In this text, he explains that the Ursprache, our "original" language, is "blissful" precisely because it lacks the arbitrariness that results from overdetermination.  Ur-speech is Adamic language, the linguistic power that God gives to Adam to confer identity on the material world.  It contains no arbitrary component, but reveals the unity between God's divine plan and the world as it exists.  Before ruins and fragments, there is no overdetermination to induce the melancholy of acedia.  Instead the originary language implies a unity of transcendent and immanent realms.  "With the creative omnipotence of language it begins, and at the end of language, as it were, assimilates the created, names it.  Language is therefore both the creative and the finished creation; it is word and nature."6

This blissful state between the world and its creator as expressed in Adamic language has its end, of course, in the Fall.  The "ignorance" introduced into the world that ultimately drives our melancholic state of acedia has its inception with the Fall away from the edenic union that joins God's plan to the immediacy of the material world.  What ensues, says Benjamin, is an overabundance of conventional languages, a prattle of meanings now localized hence arbitrary.  A former connection to a defining origin has been lost; and an overdetermined, plethoric state of melancholia forms.  Over-determination stems from over-naming.  "Things have no proper names except in God.  . . . In the language of men, however, they are overnamed."  Overnaming becomes "the linguistic being of melancholy."7

      6 Walter Benjamin, "On Language as Such and On the Languages of Man," Edmund Jephcott, tr., Walter Benjamin, Selected Writings, Volume I:  1913-1926, Marcus Bullock and Michael W. Jennings, eds., Cambridge, MA, Harvard University Press, 1997, p. 68.  
      7 Ibid., p. 73.

For a Christian perspective on Adamic language, see Charles Williams's The Place of the Lion.

 

See also the previous entry:

Float like a butterfly,
sting like a

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The image “http://www.log24.com/log/pix06A/060602-Winner.jpg” cannot be displayed, because it contains errors.

Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 pm
Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares
(ps),
a discussion of a 4×4 array
of square linear designs.
  Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle:  the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16.  The first group is the most obvious.  The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
  Being and Nothingness,
  Philosophical Library, 1956
  [reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
 

  The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1. on Page 51:
“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being.  Man is ‘a being of distances.'”
2. on Page 154:
“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities.  Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3. on Page 155:
“… end of the desires.  But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the  glass-in-the-midst-of-the-world.”
4. on Page 158:
“…  it is in time that my possibilities appear on the horizon of the world which they make mine.  If, then, human reality is itself apprehended as temporal….”
5. on Page 180:
“… else time is an illusion and chronology disguises a strictly logical order of  deducibility.  If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6. on Page 186:
“…  It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7. on Page 332:
“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon.  It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8. on Page 359:
“… eyes as objects which manifest the look.  The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9. on Page 392:
“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.  on Page 502:
“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”
11.  on Page 506:
“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.  on Page 661:
“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.  on Page 754:
“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself.  The pen and the pipe, the clothing, the desk, the house– are myself.  The totality of my possessions reflects the totality of my being.  I am what I have.  It is I myself which I touch in this cup, in this trinket.  This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

The image “http://www.log24.com/log/pix05/050527-CipraLogo.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050527-CIPRAview.jpg” cannot be displayed, because it contains errors.
 
From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

Tuesday, October 12, 2004

Tuesday October 12, 2004

Filed under: General — Tags: , — m759 @ 4:30 pm

The Last Enemy
(See April 30)

  

"I was also impressed… by the intensity of Continental modes of literary-critical thought….

On the Continent, studies of Hölderlin and Rousseau, of Poe, Baudelaire, Mallarmé and Rilke, of Rabelais, Nietzsche, Kafka, and Joyce, challenged not only received ideas on the unity of the work of art but many aspects of western thought itself. Derrida, at the same time, who for nearly a decade found a home in Yale's Comparative Literature Department, expanded the concept of textuality to the point where nothing could be demarcated as 'hors d'œuvre' and escape the literary-critical eye. It was uncanny to feel hierarchic boundaries waver until the commentary entered the text—not literally, of course, but in the sense that the over-objectified work became a reflection on its own status, its stability as an object of cognition. The well-wrought urn contained mortal ashes."

— Geoffrey Hartman, A Life of Learning

In memory of
Jacques Derrida and James Chace,
both of whom died in Paris on
Friday, Oct. 8, 2004… continued…
(See previous three entries.)

The image “http://www.log24.com/log/pix04A/041010-Welles.gif” cannot be displayed, because it contains errors.

Orson Welles


Mate in 2
V. Nabokov, 1919

"The last enemy
that shall be destroyed is death."
— Saul of Tarsus, 1 Cor. 15:26

The image “http://www.log24.com/log/pix04A/041012-Welles.gif” cannot be displayed, because it contains errors.

Knight move,
courtesy of V. Nabokov:

Nfe5 mate

The image “http://www.log24.com/log/pix04A/041012-Kt.gif” cannot be displayed, because it contains errors.

Knight:

Sir John Falstaff
(See Chimes at Midnight.)

Friday, September 24, 2004

Friday September 24, 2004

Filed under: General,Geometry — Tags: — m759 @ 6:49 pm

Readings for
Yom Kippur

The film Pi is, in part, about an alleged secret name of God that can be uttered only on Yom Kippur.  This is my personal version of such a name– not an utterance, but instead a picture:

6:49:32 PM
Sept. 24, 2004

Complete graph K6

The Details:

 

Sylvester's Music 

The Unity of Mathematics

720 in the Book

Synthemes and Spreads (pdf)

(Appendix A of
"Classification of
Partial Spreads in PG(4,2)
,"
by Leonard H. Soicher et al.)

Saturday, May 15, 2004

Saturday May 15, 2004

Filed under: General — m759 @ 4:23 am

Language Game

In memory of
Samuel Iwry, Hebrew scholar,
who died on May 8, 2004:

From a log24 entry of May 8, 2004,
on Wittgenstein’s “language games” —

    “Let us imagine a language …”

— Ludwig Wittgenstein,
    Philosophical Investigations

Okay…

Moral of the story:
If you must have a
religious language,
Elvish may,
in some situations,
do as well as Hebrew.

See also

The Unity of Mathematics,
or Shema, Israel
.

Tuesday, January 6, 2004

Tuesday January 6, 2004

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

720 in the Book

Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.

An article titled On Mathematical Imagination concludes by looking forward to

“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”

Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.

Hmm.

The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:

Abel’s Proof:  An Essay
on the Sources and Meaning
of Mathematical Unsolvability

by Peter Pesic,
MIT Press, 2003

From a review:

“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….

Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations.  The reader is left with little clarity on this sequel to the story….”

— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244

Here, it seems, is my epiphany:

“Elliptic modular functions suffice to solve all polynomial equations.”


Incidental Remarks
on Synchronicity,
Part I

Those who seek a star
on this Feast of the Epiphany
may click here.


Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.

Just how such equations can be solved is a less familiar story.  I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.

The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions.  Modular functions are also distantly related, via the topic of “moonshine” and via the  “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.


Incidental Remarks
on Synchronicity,
Part II

There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.


Here is what I was able to find on the Web about Pesic’s claim:

From Wolfram Research:

From Solving the Quintic —

“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”

From Siegel Theta Function —

“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”

From Polynomial

“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable.  Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron.  Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”

Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.

King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.

Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999. 

From Angel Zhivkov,

Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:

“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker:  in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function….  Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist.  This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions.  In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”  

— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants


Incidental Remarks
on Synchronicity,
Part III

From Music for Dunne’s Wake:

Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”

— Carrie Fisher,
Postcards from the Edge

     

720 in  
the Book”

and
Paradise

“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.

Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.

For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see

Log24, June 2003.

For the relevance of the time
of this entry, 10:10, see

  1. the reference to Paradise
    on the “postcard” above, and
  2. Storyline (10/10, 2003).

Related recreational reading:

Labyrinth



The Shining

Shining Forth

Monday, September 8, 2003

Monday September 8, 2003

Filed under: General — m759 @ 4:24 am

ART WARS Sept. 1, 2003:

Sir Terry Frost Dies

A noted English abstract painter died at 87 on Monday, September 1.  From a memorial essay on Sir Terry Frost, born in 1915, in The Daily Telegraph: 

“He was educated at Leamington Spa Central School where he edited the art magazine, but left at 15 to work….” His first jobs included, the Telegraph says, painting “the red, white and blue targets on to fighter planes.”

The “target” the Telegraph refers to
is known as the Royal Air Force Roundel.

It may indeed have functioned as a target, but it was originally intended only as a distinctive identifying mark.

Some of Frost’s later work may be viewed at the British Government Art Collection.  For some of Frost’s work more closely related to his early “target” theme, see the Badcock’s Gallery site.

An example:

 

For related religious
and cinematic material, see

Pilate, Truth, and Friday the Thirteenth,

a meditation for Good Friday of 2001,

A Matter of Life and Death,

a meditation for Friday the Thirteenth
of September, 2002,

and

The Unity of Mathematics,

from the day Frost died, which concludes
with links related to the religious symbol of

2001:

Monolithic Form
and
ART WARS.

Saturday, September 6, 2003

Saturday September 6, 2003

Filed under: General — m759 @ 2:56 pm

Pictures for Kurosawa

Five years ago on this day, director Akira Kurosawa died.

The above pictures are offered as a remembrance of Kurosawa and also of Charlotte Selver, who died on August 22, 2003.

The picture at right is from an entry of August 22.  As one obituary of Selver says, “She was very sharp and very precise.”

The picture at left is the cover of Alan Watts’s book The Spirit of Zen (a religion that is also very sharp and very precise).

From
A New Seeing,
by Mary Alice Roche

The connection with Alan Watts was a fateful one. As Charlotte recalls it, “My aunt wrote me from San Francisco, ‘last night I heard a man lecture about what you do.’ And she sent me Alan Watts’s first little book, The Spirit of Zen. I had never heard of Zen, was amazed and fascinated, and decided to visit the author.” She did so in August of 1953, and that was the beginning of a long relationship with Zen Buddhism – and also the beginning of a long series of joint seminars with Alan Watts, first in New York, and later, on Watts’s ferryboat in Sausalito, California. Some of the titles of their seminars were “Moving Stillness,” “The Unity of Opposites,” “Our Instantaneous Life,” “The Mystery of Perception,” “The Tao in Rest and Motion.” (Watts always said that Charlotte Selver taught a Western equivalent of Taoism.)

The picture at right above is intended as a sangaku, or Japanese temple tablet.

The picture at left above on the cover of Watts’s book may be regarded as illustrating the following:

“As these flowing rivers that go towards the ocean, when they have reached the ocean, sink into it, their name and form are broken, and people speak of the ocean only, exactly thus these sixteen parts of the spectator that go towards the person (purusha), when they have reached the person, sink into him, their name and form are broken, and people speak of the person only, and he becomes without parts and immortal. On this there is this verse:

‘That person who is to be known, he in whom these parts rest, like spokes in the nave of a wheel, you know him, lest death should hurt you.’ “

Prasna Upanishad

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 pm

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Tuesday, July 1, 2003

Tuesday July 1, 2003

Filed under: General — Tags: — m759 @ 5:37 pm

Jew’s on First

This entry is dedicated to those worshippers of Allah who have at one time or another cried
Itbah al-Yahud!” … Kill the Jew!
(See June 29 entries).

Dead at 78

Comedian Buddy Hackett died on Tuesday, July First, 2003, according to the New York Times.  According to Bloomberg.com, he died Sunday or Monday.

Associated Press

Buddy Hackett,
on the set of
“It’s a Mad, Mad,
Mad, Mad
World”
in 1962.

Whatever.  We may imagine he has now walked, leading a parade of many other stand-up saints, into a bar.


Hepburn at Chaillot

MIDRASH
for Buddy Hackett

From my May 25 entry,

Matrix of the Death God:

R. M. Abraham’s Diversions and Pastimes, published by Constable and Company, London, in 1933, has the following magic square:

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web include:

On Linguistic Creation

Picasso’s Birthday

1991 Yearbook
Rolling Stone



Hackett

In the Picasso’s Birthday version, 22 of the 25 magic square cells are correlated with pictures on the “Class of ’91” cover of Rolling Stone magazine.  Number 7 is Rod Stewart.  In accordance with the theological rhyme “Seven is heaven, eight is a gate,” our site music for today is “Forever Young,” a tune made famous by Stewart.

Roderick, actually   the name of the hero in “Madwoman of Chaillot”

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — Tags: — m759 @ 7:11 pm

ART WARS

Mental Health Month, Day 25:

Matrix of the Death God

Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham.  (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.)  For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.

 

See, too, The Matrix of Abraham, illustrated below.  This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web:

On Linguistic Creation

Picasso’s Birthday

A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).

This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.

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