In memory of a public intellectual who reportedly argued in favor of
"a universal identity system" and died at 94 on May 31 —
Detail from an illustration linked to here on May 31 —
Thursday, June 1, 2023
Universal Identity
Comments Off on Universal Identity
Monday, February 17, 2020
Universal Beauty
Remarks by Rosalind Krauss in the previous post suggest a look at …
Then there is the universal beauty of oneself :
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self.
Comments Off on Universal Beauty
Thursday, January 9, 2020
Universal History (Adapted)
"One of the more fortuitous encounters of late-20th-century popular culture —
almost up there with Lennon meets McCartney and Taylor meets Burton —
took place on Labor Day 1965, at Jane Fonda’s Malibu beach house. The
actress was hosting a daylong bash at which her father, Henry’s,
generation mingled uneasily with her Hollywood hippie friends. The Byrds
played in the backyard. A young comedian-turned-film director named Mike
Nichols was approached by an improv comic-turned-itinerant writer named
Buck Henry, who asked how he was doing. Nichols dourly looked around
at all the proto-Summer of Love vibes and said, 'Here, under the shadow
of the great tree, I have found peace.'
Henry immediately recognized a sardonic East Coast kindred spirit trapped
in Lotusland . . . ."
— Ty Burr, Boston Globe staff, January 9, 2020, 10:34 AM
Comments Off on Universal History (Adapted)
Sunday, December 17, 2017
Concrete Universals
The remarks on universals in the previous post linked to the following
note by James Hillman:
James Hillman, Re-Visioning Psychology ,
Harper Collins, 1977, p. 155 —
"Myths also make concrete particulars into universals,
so that each image, name, thing in my life when
experienced mythically takes on universal sense,
and all abstract universals, the grand ideas of
human fate, are presented as concrete actions."
[See note 48.]
Note 48: Cf. P. Wheelwright's discussion of concrete universality
in The Burning Fountain (Bloomington, Ind.: Indiana University
Press, 1968), pp. 52-54.
For Wheelwright's discussion, see the following excerpts from his book:
Comments Off on Concrete Universals
Saturday, February 4, 2017
Thursday, July 7, 2016
The Curious Case of the Concrete Universal
Erica Goode in the online New York Times tonight —
"Irving Gottesman, a pioneer in the field of behavioral genetics
whose work on the role of heredity in schizophrenia helped
transform the way people thought about the origins of serious
mental illness, died on June 29 at his home in Edina, Minn., a
suburb of Minneapolis. He was 85.
His wife, Carol, said he died while taking an afternoon nap.
Although Dr. Gottesman had some health problems, she said,
his death was unexpected, and several of his colleagues said
they received emails from him earlier that day."
A note from noon (EDT) on that day, June 29, for the Church of Synchronology —
A detail from the page mentioned in the June 29 post above —
A passage related to the word "soul" discussed by Sullivan —
See as well a related biblical passage, better known at the time of Royce (ca. 1892)
than today, that would probably mean nothing to the late Dr. Gottesman.
Comments Off on The Curious Case of the Concrete Universal
Sunday, July 12, 2015
Characteristica Universalis
From Wikipedia —
"Many Leibniz scholars… seem to agree that he intended
his characteristica universalis … to be a form of …
ideographic language. This was to be based on a
rationalised version of the 'principles' of Chinese characters…."
See as well O Nine, Chinese Calligraphy, and Holy Field.
Comments Off on Characteristica Universalis
Wednesday, February 29, 2012
Universals Revisited
"Some philosophical pieces are like symphonies,
others like quartets."
— Gustav Bergmann, "Frege's Hidden Nominalism,"
The Philosophical Review ,
Vol. 67, No. 4 (Oct., 1958), pp. 437-459
See also Annals of Religion.
Comments Off on Universals Revisited
Thursday, December 29, 2011
The Poetry of Universals
Continued from All Souls Day, 2011 —
Professor Sir Michael Dummett,
born June 27 1925, died December 27 2011
See also this journal on the day of Dummett's death.
Comments Off on The Poetry of Universals
Wednesday, November 2, 2011
The Poetry of Universals
A search today, All Souls Day, for relevant learning
at All Souls College, Oxford, yields the person of
Sir Michael Dummett and the following scholarly page—
My own background is in mathematics rather than philosophy.
From a mathematical point of view, the cells discussed above
seem related to some "universals" in an example of Quine.
In Quine's example,* universals are certain equivalence classes
(those with the "same shape") of a family of figures
(33 convex regions) selected from the 28 = 256 subsets
of an eight-element set of plane regions.
A smaller structure, closer to Wright's concerns above,
is a universe of 24 = 16 subsets of a 4-element set.
The number of elements in this universe of Concepts coincides,
as it happens, with the number obtained by multiplying out
the title of T. S. Eliot's Four Quartets .
For a discussion of functions that map "cells" of the sort Wright
discusses— in the quartets example, four equivalence classes,
each with four elements, that partition the 16-element universe—
onto a four-element set, see Poetry's Bones.
For some philosophical background to the Wright passage
above, see "The Concept Horse," by Harold W. Noonan—
Chapter 9, pages 155-176, in Universals, Concepts, and Qualities ,
edited by P. F. Strawson and Arindam Chakrabarti,
Ashgate Publishing, 2006.
For a different approach to that concept, see Devil's Night, 2011.
* Admittedly artificial. See From a Logical Point of View , IV, 3
Comments Off on The Poetry of Universals
Sunday, January 2, 2011
A Universal Form
Simon Critchley today in the New York Times series "The Stone"—
Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”
Actually, that's two diagonals. See Kulturkampf at the Times and Geometry of the I Ching .
Comments Off on A Universal Form
Thursday, November 5, 2009
Universal Culture Machines
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 —
and Sesame Street —
Google logo today, Nov. 5, 2009
Click images for historical background.
Comments Off on Universal Culture Machines
Friday, February 9, 2024
Sacerdotal Jargon
|
“Isn’t your work—our work—all about accessing and deploying underlying sequences and patterns? Mapping particulars on to great universals? Isn’t that the art to which, in one way or another, we’ve both devoted our best years?”
— McCarthy, Tom. The Making of Incarnation: A Novel
The hardcover first edition was published by Knopf |
|
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, |
Comments Off on Sacerdotal Jargon
Monday, December 4, 2023
Latin and Latin Squares
"… it is not just its beauty that has made Mathematics so attractive.
Thirty or so years ago, a philosopher friend of mine remarked
rather dolefully, 'I am afraid that Latin, the knowledge of which
used to be the mark of a civilised person, will be replaced by
Mathematics as the universally accepted mark of learning.'
This was probably the most prescient statement he ever made,
as the importance of Mathematics is now recognised in fields
as diverse as medicine, linguistics, and even literature."
— Address by mathematician Dominic Welsh on June 16, 2006
Some Latin-square images from pure mathematics —
Some related Latin from this journal on June 16, 2006 —
AD PULCHRITUDINEM TRIA REQUIRUNTUR:
INTEGRITAS, CONSONANTIA, CLARITAS.
INTEGRITAS, CONSONANTIA, CLARITAS.
For some remarks on Latin-square structure,
see other posts tagged Affine Squares.
Comments Off on Latin and Latin Squares
Sunday, October 1, 2023
The Usual Suspects: Schmeikal
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] |
Comments Off on The Usual Suspects: Schmeikal
Friday, August 4, 2023
The Bridge
"Finite groups of the same order are sometimes
related by a nontrivial identity."
— Steven H. Cullinane,
Groups related by a nontrivial identity
(a note on universal algebra).
See as well a related note written 40 years ago today.
A related webpage —
Comments Off on The Bridge
Wednesday, July 26, 2023
The Abstracting Which
"His is a language that is highly charged with
a harmonic resonance and a certain distancing
and abstracting which makes the reference
more universal, less specifically personal."
— Durham University, 1999 Ph.D. thesis
As verse . . .
A harmonic resonance
And a certain distancing
And abstracting which
Comments Off on The Abstracting Which
Tuesday, July 25, 2023
Eliot Illustrated
T.S. Eliot and
the music of poetry
Durham theses, Durham University, 1999
"Taking into consideration the Symbolist influence,
together with his preoccupation with language
and his interest in the musical quality inherent
in verse, one finds that Eliot's verse contains
a rhythmic movement that tends to sweep across
the whole line and links lines and stanzas together.
His is a language that is highly charged with
a harmonic resonance and a certain distancing
and abstracting which makes the reference
more universal, less specifically personal."
"Where past and future are gathered" — T. S. Eliot
Comments Off on Eliot Illustrated
Thursday, April 27, 2023
24 Views*
"Many of these same characters wonder whether
they are creating these special places anew, or
are merely finding places which already exist
(very much like 'the problem of universals' in
classical metaphysics)."
— Wikipedia article on author Roger Zelazny
Related material —
* For the title, see a story by Zelazny.
Comments Off on 24 Views*
Wednesday, March 8, 2023
Releasing the Crimson Kraken
From Chomsky's remarks in The New York Times today —
"It is at once comic and tragic, as Borges might have noted,
that so much money and attention should be concentrated
on so little a thing — something so trivial when contrasted
with the human mind, which by dint of language, in the
words of Wilhelm von Humboldt, can make 'infinite use of
finite means,' creating ideas and theories with universal reach."
A search in this journal for Humboldt University yields . . .
"Cum grano salis" — Boris Karloff in "The Black Cat."
Comments Off on Releasing the Crimson Kraken
Monday, August 1, 2022
Review
From Log24 posts tagged Art Space —
From a paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader,
and Bernd Sturmfels —
Two such considerations —
Comments Off on Review
Sunday, April 24, 2022
Structuralism: Three Betweens
|
Tuesday, November 3, 2009 Filed under: General,Geometry — Tags: Glasperlenspiel, Solomon Marcus — m759 @ 10:10 PM Book review by Jadran Mimica in Oceania, Vol. 74, 2003: "In his classic essay of 1955 'The Structural Study of Myth' Levi-Strauss came up with a universal formula of mythopoeic dynamics [fx(a) : fy(b) :: fx(b) : fa-1(y)] that he called canonical 'for it can represent any mythic transformation'. This formulation received its consummation in the four massive Mythologiques volumes, the last of which crystallises the fundamental dialectics of mythopoeic thought: that there is 'one myth only' and the primal ground of this 'one' is 'nothing'. The elucidation of the generative matrix of the myth-work is thus completed as is the self-totalisation of both the thinker and his object." So there. At least one mathematician has claimed that the Levi-Strauss formula makes sense. (Jack Morava, arXiv pdf, 2003.) I prefer the earlier (1943) remarks of Hermann Hesse on transformations of myth: "…in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created." |
Comments Off on Structuralism: Three Betweens
Saturday, April 23, 2022
Kaleidoscopic Structuralism
The previous post suggests two quotes by Elizabeth Janeway
from her review of the second volume of The Human Predicament ,
an unfinished trilogy by Richard Hughes.
"The Human Predicament poses a universal question, and Hughes
is grappling with it really as a structuralist philosopher."
"Hughes's style is kaleidoscopic , the shaking of vivid moments together
until a pattern emerges."
— The New York Times Book Review , Sunday, August 19, 1973, page 2
For a less literary example of kaleidoscopic structuralism, see
a Log24 post from the first anniversary of Janeway's reported death.
Related vocabulary —
Comments Off on Kaleidoscopic Structuralism
Saturday, April 16, 2022
The Quantum Tummelplatz
For the De las Cuevas above, see …
https://www.gemmadelascuevas.com/ —
"I am an Assistant Professor at the Institute for Theoretical Physics
at the University of Innsbruck (Austria) . . . ."
— and a tweet from Thursday, April 14, 2022, that indicates
an interest in philosophy as well as physics —
Related vocabulary —
Related drama —
Comments Off on The Quantum Tummelplatz
Friday, March 4, 2022
Popper vs. Topper
"… in the Logic of Scientific Discovery Popper introduces
the technical concept of a 'basic statement' or 'basic proposition,'
which he defines as a statement which can serve as a premise
in an empirical falsification and which takes the singular existential
form 'There is an X at Y .' Basic statements are important because
they can formally contradict universal statements, and accordingly
play the role of potential falsifiers."
— https://plato.stanford.edu/entries/popper/
Comments Off on Popper vs. Topper
Saturday, November 27, 2021
Cable
"SS refers to SuperSpeed,
a new transfer rate…"
And then there is USB,
the Universal Serial Bus . . .
From a post of 11/11, 2003.
Comments Off on Cable
Sunday, September 5, 2021
Linguistics
A Letterman introduction for Plato's Academy Awards:
"Cunning, Anna. Anna, Cunning." (Rimshot.)
But seriously . . .
"This work [of Wierzbicka and colleagues] has led to
a set of highly concrete proposals about a hypothesized
irreducible core of all human languages. This universal core
is believed to have a fully ‘language-like’ character in the sense
that it consists of a lexicon of semantic primitives together with
a syntax governing how the primitives can be combined
(Goddard, 1998)." — Wikipedia, Semantic Primes
Goddard C. (1998) — Bad arguments against semantic primitives.
Theoretical Linguistics 24:129-156.
Related fiction . . . Lexicon , by Max Barry (2013). See Barry in this journal.
Comments Off on Linguistics
Friday, August 20, 2021
Space Note
"Consider the six-dimensional vector space ( 𝔽2 )6
over the two-element field 𝔽2 ."
— Page 23 of "The Universal Kummer Threefold,"
arXiv:1208.1229v3, 12 June 2013, by Qingchun Ren,
Steven V. Sam, Gus Schrader, and Bernd Sturmfels.
An illustration of that space from 1981 —
The above recollection of the Kummer Threefold remark was suggested by
recent posts now tagged Smallfield . . .
|
"Third Man – an elderly American railway bum, "Art to which I fix my celebrated signature." — "Third Man" in Victor Snaith's play "Changing Stations"
|
If we read the above "art" as a scythe blade to which the "signature" —
Snaith ("the crooked handle or shaft of a scythe") — is attached,
an image of the late art critic Robert Hughes comes to mind:
That image of Hughes appeared here in a post of June 17, 2015 —
"Slow Art, Continued" — that also referenced the Kummer Threefold
paper above.
Comments Off on Space Note
Thursday, August 12, 2021
A Square Crystal Paperweight
Friday March 31, 2006
|
Comments Off on A Square Crystal Paperweight
Tuesday, August 10, 2021
Ex Fano Apollinis
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
.jpg)
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
Comments Off on Ex Fano Apollinis
Tuesday, April 6, 2021
Otro Idioma
See as well this journal on the above Sundance photo date —
“We keep coming back and coming back
To the real: to the hotel instead of the hymns . . . .”
— Wallace Stevens, quoted in posts tagged Portal1937
Update of 12:35 PM ET the same day —
Comments Off on Otro Idioma
Wednesday, March 3, 2021
Ink
From this journal on Nov. 9-12, 2004:
|
Fade to Black “…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent.” — Trevanian, Shibumi “‘Haven’t there been splendidly elegant colors in Japan since ancient times?’ ‘Even black has various subtle shades,’ Sosuke nodded.” — Yasunari Kawabata, The Old Capital  Ad Reinhardt An Ad Reinhardt painting described in the entry of
Ad Reinhardt, Abstract Painting, 1960-66. The viewer may need to tilt the screen to see that “The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.  Greek Cross There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.” — “Grids,” by Rosalind Krauss, |
Related material from The New York Times today —
Comments Off on Ink
Tuesday, September 1, 2020
Story Space
“The universe, then, is less intimation
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events.”
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events.”
– Betty Jay, reader’s guide to A Passage to India
Comments Off on Story Space
Monday, February 17, 2020
RIP Charles Portis
See also "True Grid " in this journal.
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
See as well . . .
Comments Off on RIP Charles Portis
Sunday, January 26, 2020
Duke Blocks
The Wall Street Journal Jan. 24 on a Duke University professor —
"Dr. Daubechies is best known for her work on mathematical structures
called wavelets; her discoveries have been so influential, in fact, that
these are referred to in the field as Daubechies wavelets. She describes
them as 'mathematical building blocks' that can be used to extract the
essential elements of images or signals without losing their quality—
in effect, a new universal language for scientists and researchers."
See also this journal on January 20-21, and …
Comments Off on Duke Blocks
Wednesday, November 27, 2019
Business Models
Yesterday's post on a recent sci-fi film suggests a look at LA news . . .
|
From the LA Times Monday morning — “He was working at the Children’s Television Workshop, as the treasurer or something, and I felt that wasn’t an important enough job for him,” Fuchs said. “At that time, we were doing a lot of acquisitions so he was buying music and concerts from around the world. I once asked him how he liked it, and Frank said: ‘I don’t know. There are no answers in this business.’” Biondi is credited with helping establish the successful model of a premium subscription channel…. |
As opposed to an unsuccessful model —
See also "High Life" (from a post of April 1 this year) —
Comments Off on Business Models
Monday, November 4, 2019
Trudeau Revisited
Previously in Log24: Trudeau and the Story Theory of Truth.
More-recent remarks by Trudeau —
|
Bible Stories for Skeptics
Review
About the Author
Product details |
Log24 on the above publication date — July 6, 2014 —
Comments Off on Trudeau Revisited
Monday, August 5, 2019
The Structure of Nada
|
“What did he fear? It was not a fear or dread, It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada y pues nada y nada y pues nada. Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee. He smiled and stood before a bar with a shining steam pressure coffee machine.”
— From Ernest Hemingway, |
|
Sanskrit (transliterated) —
nada:
“So Nada Brahma means not only: God the Creator
— Joachim-Ernst Berendt, |
Grace under Pressure meets Phonons under Strain .
Comments Off on The Structure of Nada
Thursday, May 16, 2019
That One Time
The title refers to an Instagram picture posted today.
Some background . . .
"It was a bright cold day in April . . . ." — George Orwell
Comments Off on That One Time
Saturday, March 9, 2019
Weapons of Mass Distraction
"Back to the Future" and . . .
I prefer another presentation from the above
Universal Pictures date — June 28, 2018 —
Comments Off on Weapons of Mass Distraction
Friday, March 8, 2019
Photo Opportunity
"I need a photo opportunity . . . ." — Paul Simon
Comments Off on Photo Opportunity
Wednesday, January 16, 2019
The Dreaming Polyhedron
"Here is a recipe for preparing a copy of the Mathieu group M24.
The main ingredient is a genus-3 regular polyhedron X
with 56 triangular faces, 84 edges, and 24 vertices.
The most delicate part of this recipe is to hold the polyhedron
by the 24 vertices and immerse the rest of it in 3-dimensional space."
— "How to Make the Mathieu Group M24 ," undated webpage
by David A. Richter, Western Michigan University
Illustration from that page —
"Another model of the (universal cover of the) polyhedron X"
Related fiction —
Cover of a 1971 British paperback edition of The Dreaming Jewels,
a story by Theodore Sturgeon (first version published in 1950):
Discuss Richter's model and the Sturgeon tale
in the context of posts tagged Aitchison.
Comments Off on The Dreaming Polyhedron
Monday, August 13, 2018
Trojan Horsitude
|
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, |
". . . a perfect triptych of horsitude"
— James Parker on the 2007 film "Michael Clayton"
Related material —
Horsitude in the 4×2 grid, and
Comments Off on Trojan Horsitude
Saturday, June 16, 2018
Kummer’s (16, 6) (on 6/16)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 166 Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
Comments Off on Kummer’s (16, 6) (on 6/16)
Thursday, June 7, 2018
Paved with Good Intentions
See also David Brooks on logic in today’s online New York Times —
“…the necessary skill of public life, the ability to
see two contradictory truths at the same time.”
Comments Off on Paved with Good Intentions
Monday, May 7, 2018
Fish Babel
Stanley Fish in the online New York Times today —
". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.
Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"
The final page, 759, of the Harry Potter saga —
"Talk about magical thinking!" — Fish, ibidem .
See also the above Harry Potter page
in this journal Sunday morning.
Comments Off on Fish Babel
Saturday, December 16, 2017
Dagger Definitions (Review)
The previous post suggests a review of
the philosophical concept of universals —
A part of the above-mentioned 2011 "Saturday evening's post" that is
relevant to the illustration at the end of today's previous post —
Note the whatness of Singer's dagger definitions —
Comments Off on Dagger Definitions (Review)
Saturday, September 2, 2017
A Touchstone
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
Comments Off on A Touchstone
Monday, July 31, 2017
Stealth Cars
From a June 8, 1987, New York Times review of
Stephen King's novel Misery —
"She doesn't like Fast Cars , the manuscript of which she found
in his traveling bag. It's confusing and the language is profane."
From the IMDb biography of film director Rob Cohen —
"He attended Harvard University and graduated
magna cum laude in the class of '71, concentrating
in a cross major between anthropology and visual studies."
"He is the creator of The Fast and the Furious (2001),
Universal Pictures' biggest franchise of all time."
Cohen also directed Stealth (2005). See a Sam Shepard fan site.
Comments Off on Stealth Cars
Monday, June 26, 2017
Four Dots
Analogies — “A : B :: C : D” may be read “A is to B as C is to D.”
Gian-Carlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
See also Four Dots in this journal.
Some context: McLuhan + Analogy.
Comments Off on Four Dots
Thursday, April 27, 2017
Road to Hell
An image in the previous post referred to something called
“universal logic,” touted in 2015 by the publisher Birkhäuser*
as a “new interdisciplinary field.”
From this journal on April 20 last year —
Universal Logic and the Road to Hell.
* See the webpage excerpted below.
Comments Off on Road to Hell
Wednesday, March 29, 2017
Art Space, Continued
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
Comments Off on Art Space, Continued
Tuesday, February 21, 2017
Zen and the Art
From today's print version of the New York Times —
“He eliminates anything that’s not essential
from the face of this little rabbit until it’s really
reduced to the absolute minimum,”
Mr. Dibbits said. “And he does the same for
the text of his children’s books. He uses a
language that’s not simple or stupid, but he
reduces to the bare essentials.”
About his own work, Mr. Bruna once said,
“I spend a long time making my drawings
as simple as possible, throwing lots away,
before I reach that moment of recognition.”
He added, “I leave plenty of space for children’s
imagination.”
The result is a series of “Zen-like” tales,
Ms. Vogt said, “and that’s also part of the
universal appeal.”
The passage above is from an obituary for an artist who
reportedly died on Feb. 16.
See also, in this journal, "How deep the rabbit hole goes."
Comments Off on Zen and the Art
Sunday, February 5, 2017
Against Bewitchment
A footnote in memory of a preservationist —
Title page of a thesis on language by Miles Spencer Kimball from 1984
The previous post's quotation from the Kimball thesis contains
a reference (numbered 23) to the source of Wittgenstein's
"savages" remark: Philosophical Investigations , § 194.
Kimball's remarks quoted in the previous post are from
page 121 of his thesis, under the heading "Wittgenstein's
Battle Against Bewitchment by Language."
From a cinematic example of such bewitchment —
Comments Off on Against Bewitchment
Saturday, February 4, 2017
♫ Are You Going to Vanity Fair?
"In those days, the occult sciences were cultivated with ardor well calculated to surprise the incredulous minds of our own sovereignly analytical age; perhaps they may detect in this historical sketch the germ of the positive sciences, widely studied in the nineteenth century, but without the poetic grandeur which was ascribed to them by the audacious investigators of the sixteenth century; who, instead of devoting their energy to industry, magnified art and made thought fruitful. The patronage universally accorded to art by the sov- ereigns of that time was justified, too, by the mar- vellous creations of inventors who started in quest of the philosopher's stone and reached amazing re- sults." — Balzac, Catherine de' Medici
Honoré de Balzac, Sur Catherine de Médicis :
— Hé! bien, sire, en ôtant Dieu de ce monde, que reste-t-il?
L’homme! Examinons alors notre domaine?
Le monde matériel est composé d’éléments, ces éléments
ont eux-mêmes des principes. Ces principes se résolvent
en un seul qui est doué de mouvement. Le nombre TROIS est
la formule de la création: la Matière, le Mouvement, le Produit!
— La preuve? Halte-là, s’écria le roi.
Illustration by Frederick Alfred Rhead of Vanity Fair,
page 96 in the John Bunyan classic Pilgrim's Progress
(New York, The Century Co., 1912)
Comments Off on ♫ Are You Going to Vanity Fair?
Friday, February 3, 2017
Hard Kernel
Hermeneutics —
The above quote occurs in a search called up by clicking on the image
of Amy Adams in the noon post on Groundhog Day (yesterday).
For a "universal message" see the final post of Groundhog Day.
For an "unintelligible secret," see today's previous post.
See also kernel in this journal.
Comments Off on Hard Kernel
Thursday, February 2, 2017
Hollywood Easter Egg
From yesterday's post Hollywood Arrival —
" Yesterday's events at 6407 Sunset Blvd. in Hollywood,
together with the logic of number and time from recent
posts based on a Heinlein short story, suggest that the
character played by Adams is a sort of 'fifth element'
needed to save the world.
In other words, the strange logic of recent posts ties the
California lottery number 6407 to the date April 12, 2015,
and a check of that date in this journal yields posts tagged
Orthodox Easter 2015 that relate to the 'fifth element.' "
A related image from Arrival (at 1:37:18) —
A related passage of scientific prose —
"Paramount discoveries are still being made…."
Or at least distributed.
Comments Off on Hollywood Easter Egg
Monday, January 23, 2017
The In Crowd
The New York Times online on Thursday, Jan. 19, 2017 —
" 'Split' is being released by Universal under the Blumhouse label,
a brand associated with unpretentious, clever, neo-traditionalist
scare-pictures like 'Insidious,' 'Paranormal Activity' and 'The Purge.'
That seems like the right company for Mr. Shyamalan . . . ."
A check of the Blumhouse label leads to a NY Times article
dated July 15, 2012 —
Comments Off on The In Crowd
Monday, July 25, 2016
Viennese Thought Continues
"Now vee may perhaps to begin. Yes?" — Philip Roth
Comments Off on Viennese Thought Continues
Saturday, July 23, 2016
Annals of Viennese Thought
From a University of Vienna researcher
quoted in yesterday's 11 AM ET post —
For further University of Vienna meditations, see Fitch.
Comments Off on Annals of Viennese Thought
Sunday, May 8, 2016
The Three Solomons
Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.
A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."
|
From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE "The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed." "Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)" |
See also the Cullinane models of some small Galois spaces —
Comments Off on The Three Solomons
Wednesday, April 20, 2016
Symmetric Generation of a Simple Group
The reference in the previous post to the work of Guitart and
The Road to Universal Logic suggests a fiction involving
the symmetric generation of the simple group of order 168.
See The Diamond Archetype and a fictional account of the road to Hell …
The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.
For related symmetric generation of a much larger group, see Solomon’s Cube.
Comments Off on Symmetric Generation of a Simple Group
Sunday, January 24, 2016
Long Line
"The ideal of a complete mathematical theory of beauty
lies on the same long line of distinguished fantasies of
mathematical wisdom as the number mysticism of
Pythagoras and Plato, the Ars Magna of Ramon Llull
(whom Agrippa studied) and Giordano Bruno
(who studied Llull and Agrippa), the vision of Mathesis
Universalis that Descartes and Leibniz shared, and the
Ars Combinatoria of Leibniz. Dürer does not deny the
existence of absolute beauty but despairs of knowing it."
— The late David Ritz Finkelstein in 2007.
He reportedly died today.
Comments Off on Long Line
Tuesday, December 8, 2015
Conceptual Art
A December 7th New York Times column:
A current exhibition by Joseph Kosuth in Oslo:
From the two texts by Mondrian at the right hand of Kosuth —
"The positive and negative states of being bring about action."
"Through its pure relationships, purely abstract art
can approach the expression of the universal …."
These texts may be viewed as glosses on the following image —
Click image for related posts.
Comments Off on Conceptual Art
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
Comments Off on Charm School
Monday, October 19, 2015
Symmetric Generation of the Simple Order-168 Group
This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —
“Hexagonal Logic of the Field F8 as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in
The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,
Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.
See also the eightfold cube in this journal.
Comments Off on Symmetric Generation of the Simple Order-168 Group
Tuesday, October 13, 2015
The Tombstone Source
Readings from this afternoon's online New York Times :
THE SOURCE:
|
<p class="story-body-text story-content" The writer Ari Shavit has written that Ein Harod is ‘‘imprinted on every Israeli’s psyche,’’ a microcosm of the Zionist project itself. ‘‘In a sense it is our Source,’’ he writes, ‘‘our point of departure.’’ And yet the often-overlooked museum at its heart is a different kind of symbol, at once more personal and more universal. God may have been banished from Ein Harod, but there, in a humble building on a kibbutz that has seen better days, you experience the Psalmic ideal of being ‘‘enveloped in light,’’ and with it, a reminder of history’s emotional inner life. <span class="tombstone"><i class="icon"></i></span></p> |
STYLE CODE:
Comments Off on The Tombstone Source
Sunday, August 9, 2015
Nine is a Vine
Backstory: That phrase in this journal.
““The serpent’s eyes shine
As he wraps around the vine….”
– Don Henley
With Derrida, as usual, playing the role of
the serpent, see a philosophical meditation from
October 9, 2014, by a perceptive and thoughtful Eve
that includes the following passage:
“But, before this and first of all, there is
the resistance posed by the work itself,
the hard kernel formed when the intelligibility
of a universal ‘message’ is joined to the
unintelligible secret of a singularity.”
See as well the word “kernel” here.
Comments Off on Nine is a Vine
Sunday, July 12, 2015
Concept Script
"Historically, the idea of a concept script
derives from the Leibnizian project of developing
a so-called 'universal characteristic'
(characteristica universalis )…."
— Dorothea Lotter, "Gottlob Frege: Language,"
in Internet Encyclopedia of Philosophy
Frege as quoted by Lotter —
"Arithmetical, geometrical and chemical symbols
can be regarded as realizations of the Leibnizian
conception in particular fields. The concept script
offered here adds a new one to these – indeed,
the one located in the middle, adjoining all the others."
Wittgenstein —
"Philosophy is a battle against the bewitchment
of our intelligence by means of our language."
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
— Philosophical Investigations (1953), Section 109
Frege, Preface to the Begriffsschrift —
"If it is one of the tasks of philosophy
to break the domination of words over the human spirit
by laying bare the misconceptions
that through the use of language
often almost unavoidably arise
concerning the relations between concepts
and by freeing thought from that with which only
the means of expression of ordinary language,
constituted as they are, saddle it,
then my ideography, further developed for these purposes,
can become a useful tool for the philosopher."
"Wenn es eine Aufgabe der Philosophie ist,
die Herrschaft des Wortes über den menschlichen Geist
zu brechen, indem sie die Täuschungen aufdeckt,
die durch den Sprachgebrauch über die Beziehungen der Begriffe
oft fast unvermeidlich entstehen,
indem sie den Gedanken von demjenigen befreit, womit ihn allein
die Beschaffenheit des sprachlichen Ausdrucksmittels behaftet,
so wird meine Begriffsschrift, für diese Zwecke weiter ausgebildet,
den Philosophen ein brauchbares Werkzeug werden können."
Comments Off on Concept Script
Saturday, July 11, 2015
Midnight in the Garden of Allah
From Universals Revisited, Leap Day, 2012 —
Comments Off on Midnight in the Garden of Allah
Monday, June 1, 2015
Excited about Space
Background — A search for the title in this journal.
Grothendieck was at times excited about space:
"The notion of space is certainly one of the oldest
in mathematics. It is fundamental to our 'geometric'
perspective on the world, and has been so tacitly
for over two millenia. It's only over the course of the
19th century that this concept has, bit-by-bit, freed
itself from the tyranny of our immediate perceptions
(that is, one and the same as the 'space' that
surrounds us), and of its traditional theoretical
treatment (Euclidean), to attain to its present
dynamism and autonomy. In our own times it has
joined the ranks of those notions that are most freely
and universally employed in mathematics, and is
familiar, I would say, to every mathematician
without exception. It has become a concept of multiple
and varied aspects, of hundreds of thousands of faces…."
— fermentmagazine.org/rands/promenade12.html
An aspect not so familiar: Diamond Space.
Comments Off on Excited about Space
Wednesday, March 11, 2015
World Time
1:46 AM March 11, 2015 EDT
(Eastern Daylight Time in USA) is
5:46 AM March 11, 2015 UTC
(Coordinated Universal Time worldwide).
"On March 11, 2011 at 5:46 a.m.
Coordinated Universal Time (UTC),
a magnitude 8.9 earthquake occurred
off the East Coast of Japan…."
Comments Off on World Time
Tuesday, February 3, 2015
Expanding the Spielraum
A short poem by several authors:
"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."
— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
http://arxiv.org/abs/1208.1229v3,
August 6, 2012 — June 12, 2013.
See also "Expanded Field" in this journal.
Illustration from "Sunday School," July 20, 2014.
Comments Off on Expanding the Spielraum
Tuesday, September 23, 2014
Both Hands and an Ass Map
Remarks by University Diaries this morning suggested a search for
Rutgers in this journal. That search yields a post from Dec. 30, 2005,
that is closely related to both this morning's previous post and to recent
Log24 remarks on entities and concrete universals .
Comments Off on Both Hands and an Ass Map
Sunday, September 21, 2014
Saturday-Morning Concept
“Why Is Our Sci-Fi So Glum About A.I.?,”
by Jayson Greene, NY Times Sunday Magazine today —
“You come to pity these advanced beings, bumping against
the dunderheaded constraints that their less-advanced
creators have placed on them. Johansson’s Lucy grows so
powerful as her cerebral capacity multiplies that she is able
to manipulate her cellular structure. And yet, when pursued
by an entire planet’s worth of law enforcement, she settles
on a disguise straight out of Saturday-morning cartoons —
really big sunglasses and a hairdo change.”
See also this journal on Saturday morning for a definition, and
Geometry of the I Ching for examples, of …
changeable, instantiable entities, i.e., concrete universals.
Comments Off on Saturday-Morning Concept
Saturday, September 20, 2014
The Metaphysics of Entities
Anthony Lane in The New Yorker , issue dated Sept. 22, 2014:
"The hero of 'The Zero Theorem' is a computer genius called Qohen Leth
(Christoph Waltz)…. He is the sole resident of a derelict church, where,
on a crucifix in front of the altar, the head of Christ has been replaced by
a security camera. No prayers are ever said, and none are answered.
In short, the place is deconsecrated, but to claim that it lacks any spark of
sacred yearning would be wrong, because Qohen devotes his days to seeking
the Zero Theorem, which—whatever it may be—lies at the fuzzy limit of
human powers. “We crunch entities,” he says, as if that explained anything.
His employer is Mancom, a large corporation that, in Orwellian fashion,
oversees ordinary lives, although it betrays more frantic desperation than
glowering threat."
One approach to the metaphysics of entities was indicated in the previous
post, 'Metaphysics for Gilliam." A different approach:
"Categories, Sets, and the Nature of Mathematical Entities,"
by Jean-Pierre Marquis, Ch. 13, pp. 181-192, in the 2006 book
The Age of Alternative Logics , ed. by van Benthem et al.
(Springer, Netherlands).
From pages 182-183 —
13.2 The nature of mathematical entities
Let us start with the nature of mathematical entities in general and with a
rough and classical distinction that will simply set the stage for the picture we
want to develop. We essentially follow Lowe 1998* for the basic distinctions. We
need to distinguish between abstract and concrete entities, on the one hand, and
universals and particulars on the other hand. For our purpose, it is not necessary
to specify a criterion of demarcation between abstract and concrete entities. We
simply assume that such a distinction can be made, e.g. concrete entities can
change whereas abstract entities cannot. We assume that a universal is an entity
that can be instantiated by entities which themselves are not instantiable, the
latter being of course particulars. Given these distinctions, an entity can be a
concrete particular, a concrete universal, an abstract particular or an abstract
universal.
Our focus here is between the last two possibilities. For we claim that the
current conception of sets makes them abstract particulars whereas for objects
defined within categories, mathematical entities are abstract universals. This,
we claim, is true of category theory as it is.
* Lowe, E.J., 1998, The Possibility of Metaphysics , Oxford: Clarendon Press.
Comments Off on The Metaphysics of Entities
Friday, September 19, 2014
Metaphysics for Gilliam
See also…
Related remarks: Diederik Aerts at arXiv.org.
See also Aerts (as above) on the metaphysics of entities (1984):
Comments Off on Metaphysics for Gilliam
Song of Praise
“Robust and compact, conforming to international standards,
the Altivar 08 is a universal product.”
Comments Off on Song of Praise
Saturday, July 26, 2014
OOPs
Or: Two Rivets Short of a Paradigm
Detail from an author photo:
|
From rivet-rivet.net:
The philosopher Graham Harman is invested in re-thinking the autonomy of objects and is part of a movement called Object-Oriented-Philosophy (OOP). Harman wants to question the authority of the human being at the center of philosophy to allow the insertion of the inanimate into the equation. With the aim of proposing a philosophy of objects themselves, Harman puts the philosophies of Bruno Latour and Martin Heidegger in dialogue. Along these lines, Harman proposes an unconventional reading of the tool-being analysis made by Heidegger. For Harman, the term tool does not refer only to human-invented tools such as hammers or screwdrivers, but to any kind of being or thing such as a stone, dog or even a human. Further, he uses the terms objects, beings, tools and things, interchangeably, placing all on the same ontological footing. In short, there is no “outside world.” Harman distinguishes two characteristics of the tool-being: invisibility and totality. Invisibility means that an object is not simply used but is: “[an object] form(s) a cosmic infrastructure of artificial and natural and perhaps supernatural forces, power by which our last action is besieged.” For instance, nails, wooden boards and plumbing tubes do their work to keep a house “running” silently (invisibly) without being viewed or noticed. Totality means that objects do not operate alone but always in relation to other objects–the smallest nail can, for example, not be disconnected from wooden boards, the plumbing tubes or from the cement. Depending on the point of view of each entity (nail, tube, etc.) a different reality will emerge within the house. For Harman, “to refer to an object as a tool-being is not to say that it is brutally exploited as a means to an end, but only that it is torn apart by the universal duel between the silent execution of an object’s reality and the glistening aura of its tangible surface.” — From "The Action of Things," an M.A. thesis at the Center for Curatorial Studies, Bard College, by Manuela Moscoso, May 2011, edited by Sarah Demeuse |
From Wikipedia, a programming paradigm:
See also posts tagged Turing's Cathedral, and Alley Oop (Feb. 11, 2003).
Comments Off on OOPs
Friday, March 7, 2014
Kummer Varieties
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7, W (E7):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .
Comments Off on Kummer Varieties
Sunday, June 30, 2013
Book Award
|
"What on earth is
— Said to be an annotation |
In the spirit of the late Thomas Guinzburg…
See also "Concrete Universal" in this journal.
Related material— From a Bloomsday reply
to a Diamond Theory reader's comment, an excerpt—
The reader's comment suggests the following passages from
the book by Stirling quoted above—
Here Stirling plays a role analogous to that of Professor Irwin Corey
accepting the National Book Award for Gravity's Rainbow in 1974.
Comments Off on Book Award
Monday, December 24, 2012
Eternal Recreation
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195-196:
"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."
The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
Comments Off on Eternal Recreation
Tuesday, September 11, 2012
Symmetry and Hierarchy
A followup to Intelligence Test (April 2, 2012).
Philosophical Transactions of the Royal Society
B (2012) 367, 2007–2022
(theme issue of July 19, 2012) —
Gesche Westphal-Fitch [1], Ludwig Huber [2],
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]
[1] Department of Cognitive Biology, University of Vienna,
Althanstrasse 14, 1090 Vienna, Austria
Althanstrasse 14, 1090 Vienna, Austria
[2] Messerli Research Institute, University of Veterinary Medicine Vienna,
Medical University of Vienna and University of Vienna,
Veterinärplatz 1, 1210 Vienna, Austria
Medical University of Vienna and University of Vienna,
Veterinärplatz 1, 1210 Vienna, Austria
[3] School of Psychology, St Mary’s College, University of St Andrews,
South Street, St Andrews, KY16 9JP, UK
South Street, St Andrews, KY16 9JP, UK
Excerpt (added in an update on Dec. 8, 2012) —
Conclusion —
"… We believe that applying the theoretical
framework of formal language theory to two-dimensional
patterns offers a rich new perspective on the
human capacity for producing regular, hierarchically
organized structures. Such visual patterns may actually
prove more flexible than music or language for probing
the full extent of human pattern processing abilities.
With the results presented here, we have taken the
first steps in decoding the uniquely human fascination
with visual patterns, what Gombrich termed our
‘sense of order’.
Although the patterns we studied are most similar
to tilings or mosaics, they are examples of a much
broader type of abstract plane pattern, a type found
in virtually all of the world’s cultures [4]. Given that
such abstract visual patterns seem to represent
human universals, they have received astonishingly
little attention from psychologists. This neglect is particularly
unfortunate given their democratic nature,
their popular appeal and the ease with which they
can be generated and analysed in the laboratory.
With the current research, we hope to spark renewed
scientific interest in these ‘unregarded arts’, which
we believe have much to teach us about the nature of
the human mind."
the human mind."
[4] Washburn, D. K. & Crowe, D. W.,1988
Symmetries of Culture:
Theory and Practice of Plane Pattern Analysis.
Seattle, WA: University of Washington Press.
Symmetries of Culture:
Theory and Practice of Plane Pattern Analysis.
Seattle, WA: University of Washington Press.
Commentary —
For symmetry , see Finite Geometry of the Square and Cube.
For hierarchy , see my assessment of Gombrich.
For culture , see T. S. Eliot and Russell Kirk on Eliot.
For culture , see T. S. Eliot and Russell Kirk on Eliot.
Comments Off on Symmetry and Hierarchy
Tuesday, July 31, 2012
Logo
"The universe, then, is less intimation
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events."
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events."
– Betty Jay, reader's guide to A Passage to India
Judy Davis in the Marabar Caves
The above image is from this journal on Sunday, April 13, 2008.
The preceding cover of a book by Northrop Frye was suggested
by material in this journal from February 2003.
See also Yankee Puzzle and Doodle Dandy.
Comments Off on Logo
Thursday, March 1, 2012
Block That Metaphor:
The Cube Model and Peano Arithmetic
The eightfold cube model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—
The Consistency of Arithmetic, by Storrs McCall
"Is Peano arithmetic (PA) consistent? This paper contains a proof that it is. …
Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules. 'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects. If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency. But in addition to axiomatic proofs there is another kind of proof. In this paper I give a proof of PA's consistency based on a formal semantics for PA. To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.
The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :
"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory. This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models. In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations. The models occupy centre stage." (1980, p. 44)
Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model. Suppose we wish to prove the consistency of the following geometric axioms:
A1. For any two lines, there is at most one point that lies on both.
A2. For any two points, there is exactly one line that lies on both.
A3. On every line there lie at least two points.
The following diagram shows the axioms to be consistent:
Figure 1
The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure. It is a semantic as opposed to a syntactic proof. The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….
… The semantics presented in this paper I call 'block semantics', for reasons that will become clear…. Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks. These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns. A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable. Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows. The result is called the 'linear transformation' of the rectangle. An essential characteristic of block semantics is that every domain of every block model is finite. In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains. But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.
It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"
— Storrs McCall, Department of Philosophy, McGill University
See also…
- Finite Geometry and Physical Space,
- McCall in the book search of this morning's Salvation for Gopnik, and
- the post Declarations of Jan. 29, 2012.
Comments Off on Block That Metaphor:
Salvation for Gopnik
Yesterday afternoon's post Universals Revisited linked
(indirectly) to an article in the current New Yorker on
the Book of Revelation —
"The Big Reveal: Why Does the Bible End That Way?"
The connection in that post between universals and Revelation
may have eluded readers unfamiliar with a novel by
Charles Williams, The Place of the Lion (London, Gollancz, 1931).
The article's author, Adam Gopnik, appears in the following
Google Book Search, which may or may not help such readers.
Should Gopnik desire further information on Williams and salvation,
he may consult Steps Toward Salvation: An Examination of Coinherence
and Substitution in the Seven Novels of Charles Williams ,
by Dennis L. Weeks (American University Studies: Series 4, English
Language and Literature. Vol. 125), XV + 117 pp., Peter Lang Publishing, 1991.
The ninth item in the above search refers to a boxed set
of the seven novels themselves—
.
Comments Off on Salvation for Gopnik
Thursday, January 5, 2012
Precisely
From a review of Truth and Other Enigmas , a book by the late Michael Dummett—
"… two issues stand out as central, recurring as they do in many of the
essays. One issue is the set of debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent of the way we
represent that aspect to ourselves. For example, is there a realm of mathematical
entities that exists fully formed independently of our mathematical activity? Are
there facts about the past that our use of the past tense aims to capture? The other
issue is the view— which Dummett learns primarily from the later Wittgenstein—
that the meaning of an expression is fully determined by its use, by the way it
is employed by speakers. Much of his work consists in attempts to argue for this
thesis, to clarify its content and to work out its consequences. For Dummett one
of the most important consequences of the thesis concerns the realism debate and
for many other philosophers the prime importance of his work precisely consists
in this perception of a link between these two issues."
— Bernhard Weiss, pp. 104-125 in Central Works of Philosophy , Vol. 5,
ed. by John Shand, McGill-Queen's University Press, June 12, 2006
The above publication date (June 12, 2006) suggests a review of other
philosophical remarks related to that date. See …
- a June 12, 2006 link (in a 2007 post) to Geometry of the 4×4 Square,
- some related remarks on Dummett and Universals in this journal, and
- the following excerpt from Spekkens's toy model—
"Every partitioning of the set of sixteen ontic states
into four disjoint pure epistemic states
yields a maximally informative measurement."—
For some more-personal remarks on Dummett, see yesterday afternoon's
"The Stone" weblog in The New York Times.
I caught the sudden look of some dead master….
Comments Off on Precisely
Monday, December 12, 2011
X o’ Jesus
Religion for stoners,♦ in memory of Horselover Fat
Amazon.com gives the publication date of a condensed
version* of Philip K. Dick's Exegesis as Nov. 7, 2011.
The publisher gives the publication date as Nov. 8, 2011.
Here, in memory of the author, Philip K. Dick (who sometimes
called himself, in a two-part pun, "Horselover Fat"), is related
material from the above two dates in this journal—
|
Tuesday, November 8, 2011 m759 @ 12:00 PM …. Update of 9:15 PM Nov. 8, 2011— From a search for the word "Stoned" in this journal—
See also Monday's post "The X Box" with its illustration
Monday, November 7, 2011
"Design is how it works." — Steve Jobs, quoted in
For some background on this enigmatic equation,
|
Merry Xmas.
♦ See also last night's post and the last words of Steve Jobs.
* Houghton Mifflin Harcourt, the publisher, has, deliberately or not, sown confusion
about whether this is only the first of two volumes.
Comments Off on X o’ Jesus
Tuesday, November 8, 2011
Stoned*
From "The Stone" in Sunday's online New York Times—
Cosmic Imagination
By William Egginton
Do the humanities need to be defended from hard science?
Illustration of hard science —
Illustration of the humanities —
(The above illustrations from Sunday's "The Stone" are by Leif Parsons.)
Midrash by the Coen brothers— "The Dude Abides."
See also 10/10/10— The Day of the Tetractys—
* Update of 9:15 PM Nov. 8, 2011—
From a search for the word "Stoned" in this journal—
|
Sunday, January 2, 2011
m759 @ 6:40 PM Simon Critchley today in the New York Times series "The Stone"— Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”
Actually, that's two diagonals. See Kulturkampf at the Times and Geometry of the
[Here the "Stoned" found by the search |
See also Monday's post "The X Box" with its illustration
.
Comments Off on Stoned*
Thursday, November 3, 2011
Ockham’s Bubbles–
Mathematics and Narrative, continued
"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"
— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th
"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."
— Dan Conover on September 1st, 2011
"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'
I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."
— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996
"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"
— Stanford Encyclopedia of Philosophy
"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."
— Interviewer for the National Association of Scholars
From that conversation—
Wolfe : "People in academia should start insisting on objective scholarship, insisting on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"
As for "mis-imagined quantum pins"…
This journal on the date of the above interview— February 28, 2008—
Illustration from a Perimeter Institute talk given on July 20, 2005
The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by Conover, "The Last Epiphany," and four posts from September 1st, 2011—
Boundary, How It Works, For Thor's Day, and The Galois Tesseract.
Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.
"There is such a thing as a tesseract." —A Wrinkle in Time
Comments Off on Ockham’s Bubbles–
Monday, October 31, 2011
Beauty, Truth, Halloween
On Halloween…
"Remember that for Ockham there is nothing in the universe that is
in any way universal except a concept or word: there are no real
natures shared by many things. However, things do resemble one
another, some things more closely than others. So the various
degrees of resemblance give a foundation in reality for our conceptual
structures, such as Porphyry's tree.
Now resemblance (or similitude or likeness) is a relation.
If such relations are realities, then we can say that there are realities
out there that correspond to our conceptual structures."
— R.J. Kilcullen at Macquarie University, course labeled Phil360
"The kernel of a homomorphism is always a congruence.
Indeed, every congruence arises as a kernel."
— Congruence Relation, section on Universal Algebra, in Wikipedia
"Beauty then is a relation."
"An Attempt to Understand the Problem of Universals"
is the title of a talk by Fabian Geier, University of Bamberg—
"The talk was held at Gdańsk University on May 26th 2008."
Related material— Stevie Nicks turns 60.
Comments Off on Beauty, Truth, Halloween
Monday, October 24, 2011
The Lottery of Babalu
Last evening's New York Lottery numbers were 123 and 5597.
The 123 suggests page 123 of DeLillo's Underworld .
(For some context, see searches in this journal for Los Muertos and for Pearly Gates of Cyberspace .)
The 5597 suggests the birth date of literary theorist Kenneth Burke— May 5, 1897.
These two topics—
- the afterlife (in the Latin-American rhythms context of yesterday's Shine On, Edmundo)
- and Kenneth Burke
are combined in Heaven's Gate, a post from April 11, 2003—
Babylon = Bab-ilu, “gate of God,” Hebrew: Babel or Bavel.”
|
Modern rendition |
Kenneth |
The above observations on lottery hermeneutics, on a ridiculously bad translation, and on Latin rhythms did not seem worth recording until…
The New York Times Book Review for Sunday, October 30, arrived this morning.
From page 22, an extract from the opening paragraph of a review titled…
Making Sense of It
David Bellos offers a new approach to translation.
The theory of translation is very rarely— how to put this?— comical. Its mode is elegy, and severe admonishment…. You can never, so runs the elegiac argument, precisely reproduce a line of poetry in another language…. And this elegiac argument has its elegiac myth: the Tower of Babel, where the world's multiplicity of languages is seen as mankind's punishment— condemned to the howlers, the faux amis , the foreign menu apps. Whereas the ideal linguistic state would be the lost universal language of Eden.
See also Saturday's Edenville.
Comments Off on The Lottery of Babalu
Sunday, October 2, 2011
Emerson on Mathematics
From "The Poet" (1844)—
If the imagination intoxicates the poet, it is not inactive in other men. The metamorphosis excites in the beholder an emotion of joy.
The use of symbols has a certain power of emancipation and exhilaration for all men. We seem to be touched by a wand, which makes us dance and run about happily, like children. We are like persons who come out of a cave or cellar into the open air. This is the effect on us of tropes, fables, oracles, and all poetic forms. Poets are thus liberating gods. Men have really got a new sense, and found within their world, another world or nest of worlds; for the metamorphosis once seen, we divine that it does not stop. I will not now consider how much this makes the charm of algebra and the mathematics, which also have their tropes, but it is felt in every definition….
… Here is the difference betwixt the poet and the mystic, that the last nails a symbol to one sense, which was a true sense for a moment, but soon becomes old and false…. Mysticism consists in the mistake of an accidental and individual symbol for an universal one…. And the mystic must be steadily told,— All that you say is just as true without the tedious use of that symbol as with it. Let us have a little algebra, instead of this trite rhetoric,— universal signs, instead of these village symbols,— and we shall both be gainers.
See also Weyl on the use of symbols (in coordinate systems) and today's previous posts Birth of a Poet and Symmetric Generation Illustrated.
Comments Off on Emerson on Mathematics
Wednesday, August 10, 2011
Objectivity
From math16.com—
Quotations on Realism
|
The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)—
From The Search for Invariants (June 20, 2011):
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
Related material from Sunday's New York Times travel section—
Comments Off on Objectivity
Monday, August 8, 2011
Diamond Theory vs. Story Theory (continued)
Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.
(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)
From yesterday's Sunday morning New York Times—
"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."
— Drew Westen, professor at Emory University
From May 22, 2009—
|
The above ad is by
Diamond from last night’s
|
|
For further details, see Saturday's correspondences |
Comments Off on Diamond Theory vs. Story Theory (continued)
Sunday, May 22, 2011
Return of the Stone
The New York Times philosophy column "The Stone" has returned—
"There will certainly always be a place for epistemology,
or the theory of knowledge. But in order for a theory of
knowledge to tell us much, it needs to draw on examples
of knowledge of something or other." — Justin E.H. Smith
Amen.
Examples: Quine on geometry and Quine on universals.
Comments Off on Return of the Stone
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—
Comments Off on Groups Acting
Monday, April 25, 2011
The Kristen Effect
From the author of The Abacus Conundrum—
Harlan Kane's sequel to The Apollo Meme—
THE KRISTEN EFFECT
"Thus the universal mutual attraction between the sexes is represented."
— Hexagram 31
Comments Off on The Kristen Effect
Thursday, March 17, 2011
Remarks on Reality
Conclusion of “The Place of Pure Mathematics” —
“Dogmas and philosophies, it would seem, rise and fall. But gradually accumulating throughout the ages, from the earliest dawn of history, there is a body of doctrine, a reasoned insight into the relations of exact ideas, painfully won and often tested. And this remains the main heritage of man; his little beacon of light amidst the solitudes and darknesses of infinite space; or, if you prefer, like the shout of children at play together in the cultivated valleys, which continues from generation to generation.
Yes, and continues for ever! A universe which has the potentiality of becoming thus conscious of itself is not without something of which that which we call memory is but an image. Somewhere, somehow, in ways we dream not of, when you and I have merged again into the illimitable whole, when all that is material has ceased, the faculty in which we now have some share, shall surely endure; the conceptions we now dimly struggle to grasp, the joy we have in the effort, these are but part of a greater whole. Some may fear, and some may hope, that they and theirs shall not endure for ever. But he must have studied Nature in vain who does not see that our spiritual activities are inherent in the mighty process of which we are part; who can doubt of their persistence.
And, on the intellectual side, of all that is best ascertained, and surest, and most definite, of these; of all that is oldest and most universal; of all that is most fundamental and far-reaching, of these activities, Pure Mathematics is the symbol and the sum.”
— From a 1913 address by geometry saint Henry Frederick Baker, who died on this date in 1956
The feast of another saint, Patrick, also falls on 3/17. The date itself is related, if only by chance, to the following remark—
“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”
— From a 1940 book by the somewhat less saintly number theorist G. H. Hardy
Comments Off on Remarks on Reality
Wednesday, February 23, 2011
In Nomine Patris
In memory of Pete Postlethwaite, who played father figures in
"In the Name of the Father" and in "Inception."
He died on January 2, 2011.
"And I looked and behold: a pale horse" — Johnny Cash
This journal on the day of Postlethwaite's death—
For somewhat deeper discussions see "Blackburn+universals" in this journal.
"It is arguable that this is always the deepest, most profound problem of philosophy.”
– Simon Blackburn, Think (Oxford, 1999)
Comments Off on In Nomine Patris
Friday, January 14, 2011
Ironic Butterfly
David Brooks's column today quotes Niebuhr. From the same source—
Reinhold Niebuhr, The Irony of American History—
Chapter 8: The Significance of Irony
Any interpretation of historical patterns and configurations raises the question whether the patterns, which the observer discerns, are "objectively" true or are imposed upon the vast stuff of history by his imagination. History might be likened to the confusion of spots on the cards used by psychiatrists in a Rorschach test. The patient is asked to report what he sees in these spots; and he may claim to find the outlines of an elephant, butterfly or frog. The psychiatrist draws conclusions from these judgments about the state of the patient’s imagination rather than about the actual configuration of spots on the card. Are historical patterns equally subjective?
….
The Biblical view of human nature and destiny moves within the framework of irony with remarkable consistency. Adam and Eve are expelled from the Garden of Eden because the first pair allowed "the serpent" to insinuate that, if only they would defy the limits which God had set even for his most unique creature, man, they would be like God. All subsequent human actions are infected with a pretentious denial of human limits. But the actions of those who are particularly wise or mighty or righteous fall under special condemnation. The builders of the Tower of Babel are scattered by a confusion of tongues because they sought to build a tower which would reach into the heavens.
Niebuhr's ironic butterfly may be seen in the context of last
Tuesday's post Shining and of last Saturday's noon post True Grid—
The "butterfly" in the above picture is a diagram showing the 12 lines* of the Hesse configuration from True Grid.
It is also a reference to James Hillman's classical image (see Shining) of the psyche, or soul, as a butterfly.
Fanciful, yes, but this is in exact accordance with Hillman's remarks on the soul (as opposed to the spirit— see Tuesday evening's post).
The 12-line butterfly figure may be viewed as related to the discussions of archetypes and universals in Hillman's Re-Visioning Psychology and in Charles Williams's The Place of the Lion . It is a figure intended here to suggest philosophy, not entertainment.
Niebuhr and Williams, if not the more secular Hillman, might agree that those who value entertainment above all else may look forward to a future in Hell (or, if they are lucky, Purgatory). Perhaps such a future might include a medley of Bob Lind's "Elusive Butterfly" and Iron Butterfly's "In-a-Gadda-da-Vida."
* Three horizontal, three vertical, two diagonal, and four arc-shaped.
Comments Off on Ironic Butterfly
Saturday, January 8, 2011
True Grid (continued)
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the Dolgachev-Artebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
|
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
|
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
|
Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, |
And from last October —
|
Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— |
Comments Off on True Grid (continued)
Tuesday, January 4, 2011
The White Itself
The Development of Logic —
A Universal Etymological English Dictionary —
Joan Didion —
The above readings are related* to All Things Shining , a work of pop philosophy published today.
For a review of the new book, click on the image below.
* Didion's remarks on James Jones are related to the title of the new book.
Jones wrote a novel, The Thin Red Line , on which a film is based that
contains the phrase "all things shining." The phrase is not in the novel.
The authors of All Things Shining credit neither novel nor film.
Comments Off on The White Itself
Monday, January 3, 2011
Shining
For the authors of the new book All Things Shining —
See the discussions of "concrete universals" in James Hillman's Re-Visioning Psychology and in Donald Phillip Verene's Vico and Joyce —
The index to All Things Shining contains no entries for Hillman (or his mentor Jung), Verene, Joyce, Vico, or the word "universal."
It does, however, contain four references to an example of a universal —
whiteness, 161, 169-173, 175, 178
See also "whiteness" in this journal.
Comments Off on Shining
Sunday, January 2, 2011
Horseness
"Art has to reveal to us ideas, formless spiritual essences."
— A character clearly talking nonsense, from the National Library section of James Joyce's Ulysses
"Unsheathe your dagger definitions. Horseness is the whatness of allhorse."
— A thought of Stephen Dedalus in the same Ulysses section
For a representation of horseness related to Singer's dagger definitions in Saturday evening's post, see Generating the Octad Generator and Art Wars: Geometry as Conceptual Art.
More seriously, Joyce's "horseness" is related to the problem of universals. For an illuminating approach to universals from a psychological point of view, see James Hillman's Re-Visioning Psychology (Harper Collins, 1977). (See particularly pages 154-157.)
Comments Off on Horseness
Saturday, December 18, 2010
Sunday, November 21, 2010
Fast Forward
Dan Brown, a sculpture at CIA headquarters, and secret codes are discussed at the bottom of today's New York Times front page.
In this vein, here is a meditation on Religion and Time (continued from Kurt Vonnegut's birthday)—
At the end of "Three Days of the Condor," Cliff Robertson asks Robert Redford, "How do you know they'll print it?"
One possible answer: "They always print… the lottery."
The New York Lottery on Saturday, November 20, 2010: Midday 704, Evening 687.
Here 704 suggests 7/04, the Fourth of July, which in turn suggests this journal's post on that date about random numbers and universal wisdom.
Moving further up on the front page of today's New York Times—
"Their brains are rewarded not for staying on task but for jumping to the next thing…. The worry is we’re raising a generation of kids in front of screens whose brains are going to be wired differently.”— A worried professor at Harvard Medical School
For that new generation… Live from New York, it's Universal Wisdom! — In other words, 687!
Returning to the lottery hermeneutics (look it up, dudes) of July 4th, we note that a rather arcane and archaic procedure on a Windows PC keyboard— "Num Lock + Alt + 687"— produces the symbol known as a right guillemet — ». This mark, which can stand for "fast forward," may symbolize to the old Slow-Forward Generation the fears of the Harvard professor for the new Differently-Wired Generation.
Comments Off on Fast Forward
Tuesday, July 6, 2010
What “As” Is
or: Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)
“Dasein’s full existential structure is constituted by
the ‘as-structure’ or ‘well-joined structure’ of the rift-design*…”
— Gary Williams, post of January 22, 2010
Background—
Gian-Carlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
… and projective points as separating rifts—
* rift-design— Definition by Deborah Levitt—
“Rift. The stroke or rending by which a world worlds, opening both the ‘old’ world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”
— “Heidegger and the Theater of Truth,” in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998
Comments Off on What “As” Is
Sunday, July 4, 2010
Sermon for July 4
"Throughout history mystics and philosophers have sought a compact key to universal wisdom…."
— Charles H. Bennett, "On Random and Hard-to-Describe Numbers," 1979.
Bennett was writing about Gregory Chaitin's concept of an "Omega number."
(See Thursday's "Omega at Eight" and Friday's "The Girl Who Fixed the Omega.")
Here is a different Omega number — Alt 234.
Related material:
Comments Off on Sermon for July 4
Thursday, July 1, 2010
Omega at Eight
The "compact key to universal wisdom" passage in the previous post seemed
too well written to be the work of an anonymous webforum author.
Here is a slightly expanded version—
Throughout history mystics and philosophers have sought
a compact key to universal wisdom, a finite formula or text
that would provide the answer to every question. The use of
the Bible, the Koran and the I Ching for divination and the
tradition of the secret books of Hermes Trismegistus and the
medieval Jewish Cabala exemplify this belief or hope. Such
sources of universal wisdom are traditionally protected from
casual use by being difficult to find as well as difficult to un-
derstand and dangerous to use, tending to answer more quest-
ions and deeper ones than the searcher wishes to ask. The
esoteric book is, like God, simple yet undescribable. It is om-
niscient, and it transforms all who know it. The use of clas-
sical texts to foretell mundane events is considered supersti-
tious nowadays, yet in another sense science is in quest of its
own Cabala, a concise set of natural laws that would explain
all phenomena. In mathematics, where no set of axioms can
hope to prove all true statements, the goal might be a concise
axiomatization of all "interesting" true statements.
Ω is in many senses a Cabalistic number. It can be known
of through human reason, but not known. To know it in detail
one must accept its uncomputable sequence of digits on faith,
like words of a sacred text.
This is Martin Gardner's* and Charles H. Bennett's
revised version of a passage from Bennett's paper
"On Random and Hard-to-Describe Numbers," 1979.
The original passage from Bennett's paper—
Throughout history mystics and philosophers have sought a compact key to
universal wisdom, a finite formula or text which, when known and understood,
would provide the answer to every question. The Bible, the Koran, the mythical
secret books of Hermes Trismegistus, and the medieval Jewish Cabala have
been so regarded. Sources of universal wisdom are traditionally protected from
casual use by being hard to find, hard to understand when found, and dangerous
to use, tending to answer more and deeper questions than the user wishes to
ask. Like God the esoteric book is simple yet undescribable, omniscient, and
transforms all who know It. The use of classical texts to fortell [sic] mundane events
is considered superstitious nowadays, yet, in another sense, science is in quest of
its own Cabala, a concise set of natural laws which would explain all phenomena.
In mathematics, where no set of axioms can hope to prove all true statements,
the goal might be a concise axiomatization of all "interesting" true statements.
Ω is in many senses a Cabalistic number. It can be known of, but not known,
through human reason. To know it in detail, one would have to accept its un-
computable digit sequence on faith, like words of a sacred text.
The Bennett paper deals with Gregory Chaitin's concept of an "Omega Number."
I prefer the Omega of Josefine Lyche—
See also All Hallows' Eve, 2002.
* Martin Gardner's Mathematical Games column
"The Random Number Omega Bids Fair to Hold the Mysteries of the Universe,"
Scientific American, November 1979, 241(5), pp. 20–34.
The column is reprinted as "Chaitin's Omega," Ch. 21, pp. 307-319 in the
collection of Gardner's columns titled Fractal Music, Hypercards and More,
W.H. Freeman & Co., 1991
Comments Off on Omega at Eight
Wednesday, June 23, 2010
Group Theory and Philosophy
Excerpts from "The Concept of Group and the Theory of Perception,"
by Ernst Cassirer, Philosophy and Phenomenological Research,
Volume V, Number 1, September, 1944.
(Published in French in the Journal de Psychologie, 1938, pp. 368-414.)
The group-theoretical interpretation of the fundaments of geometry is,
from the standpoint of pure logic, of great importance, since it enables us to
state the problem of the "universality" of mathematical concepts in simple
and precise form and thus to disentangle it from the difficulties and ambigui-
ties with which it is beset in its usual formulation. Since the times of the
great controversies about the status of universals in the Middle Ages, logic
and psychology have always been troubled with these ambiguities….
Our foregoing reflections on the concept of group permit us to define more
precisely what is involved in, and meant by, that "rule" which renders both
geometrical and perceptual concepts universal. The rule may, in simple
and exact terms, be defined as that group of transformations with regard to
which the variation of the particular image is considered. We have seen
above that this conception operates as the constitutive principle in the con-
struction of the universe of mathematical concepts….
…Within Euclidean geometry,
a "triangle" is conceived of as a pure geometrical "essence," and this
essence is regarded as invariant with respect to that "principal group" of
spatial transformations to which Euclidean geometry refers, viz., displace-
ments, transformations by similarity. But it must always be possible to
exhibit any particular figure, chosen from this infinite class, as a concrete
and intuitively representable object. Greek mathematics could not
dispense with this requirement which is rooted in a fundamental principle
of Greek philosophy, the principle of the correlatedness of "logos" and
"eidos." It is, however, characteristic of the modern development of
mathematics, that this bond between "logos" and "eidos," which was indis-
soluble for Greek thought, has been loosened more and more, to be, in the
end, completely broken….
…This process has come to its logical
conclusion and systematic completion in the development of modern group-
theory. Geometrical figures are no longer regarded as fundamental, as
date of perception or immediate intuition. The "nature" or "essence" of a
figure is defined in terms of the operations which may be said to
generate the figure. The operations in question are, in turn, subject to
certain group conditions….
…What we
find in both cases are invariances with respect to variations undergone by
the primitive elements out of which a form is constructed. The peculiar
kind of "identity" that is attributed to apparently altogether heterogen-
eous figures in virtue of their being transformable into one another by means
of certain operations defining a group, is thus seen to exist also in the
domain of perception. This identity permits us not only to single out ele-
ments but also to grasp "structures" in perception. To the mathematical
concept of "transformability" there corresponds, in the domain of per-
ception, the concept of "transposability." The theory of the latter con-
cept has been worked out step by step and its development has gone through
various stages….
…By the acceptance of
"form" as a primitive concept, psychological theory has freed it from the
character of contingency which it possessed for its first founders. The inter-
pretation of perception as a mere mosaic of sensations, a "bundle" of simple
sense-impressions has proved untenable….
…In the domain of mathematics this state of affairs mani-
fests itself in the impossibility of searching for invariant properties of a
figure except with reference to a group. As long as there existed but one
form of geometry, i.e., as long as Euclidean geometry was considered as the
geometry kat' exochen this fact was somehow concealed. It was possible
to assume implicitly the principal group of spatial transformations that lies
at the basis of Euclidean geometry. With the advent of non-Euclidean
geometries, however, it became indispensable to have a complete and sys-
tematic survey of the different "geometries," i.e., the different theories of
invariancy that result from the choice of certain groups of transformation.
This is the task which F. Klein set to himself and which he brought to a
certain logical fulfillment in his Vergleichende Untersuchungen ueber neuere
geometrische Forschungen….
…Without discrimination between the
accidental and the substantial, the transitory and the permanent, there
would be no constitution of an objective reality.
This process, unceasingly operative in perception and, so to speak, ex-
pressing the inner dynamics of the latter, seems to have come to final per-
fection, when we go beyond perception to enter into the domain of pure
thought. For the logical advantage and peculiar privilege of the pure con –
cept seems to consist in the replacement of fluctuating perception by some-
thing precise and exactly determined. The pure concept does not lose
itself in the flux of appearances; it tends from "becoming" toward "being,"
from dynamics toward statics. In this achievement philosophers have
ever seen the genuine meaning and value of geometry. When Plato re-
gards geometry as the prerequisite to philosophical knowledge, it is because
geometry alone renders accessible the realm of things eternal; tou gar aei
ontos he geometrike gnosis estin. Can there be degrees or levels of objec-
tive knowledge in this realm of eternal being, or does not rather knowledge
attain here an absolute maximum? Ancient geometry cannot but answer
in the affirmative to this question. For ancient geometry, in the classical
form it received from Euclid, there was such a maximum, a non plus ultra.
But modern group theory thinking has brought about a remarkable change
In this matter. Group theory is far from challenging the truth of Euclidean
metrical geometry, but it does challenge its claim to definitiveness. Each
geometry is considered as a theory of invariants of a certain group; the
groups themselves may be classified in the order of increasing generality.
The "principal group" of transformations which underlies Euclidean geome-
try permits us to establish a number of properties that are invariant with
respect to the transformations in question. But when we pass from this
"principal group" to another, by including, for example, affinitive and pro-
jective transformations, all that we had established thus far and which,
from the point of view of Euclidean geometry, looked like a definitive result
and a consolidated achievement, becomes fluctuating again. With every
extension of the principal group, some of the properties that we had taken
for invariant are lost. We come to other properties that may be hierar-
chically arranged. Many differences that are considered as essential
within ordinary metrical geometry, may now prove "accidental." With
reference to the new group-principle they appear as "unessential" modifica-
tions….
… From the point of view of modern geometrical systematization,
geometrical judgments, however "true" in themselves, are nevertheless not
all of them equally "essential" and necessary. Modern geometry
endeavors to attain progressively to more and more fundamental strata of
spatial determination. The depth of these strata depends upon the com-
prehensiveness of the concept of group; it is proportional to the strictness of
the conditions that must be satisfied by the invariance that is a universal
postulate with respect to geometrical entities. Thus the objective truth
and structure of space cannot be apprehended at a single glance, but have to
be progressively discovered and established. If geometrical thought is to
achieve this discovery, the conceptual means that it employs must become
more and more universal….
Comments Off on Group Theory and Philosophy
Sunday, June 13, 2010
Sunday School
“What on earth is a 'concrete universal'?"
— Said to be an annotation (undated)
by Robert M. Pirsig of A History of Philosophy,
by Frederick Copleston, Society of Jesus.
From Aaron Urbanczyk's 2005 review of Christ and Apollo by William Lynch, S.J., a book first published in 1960—
"Lynch's use of analogy vis-a-vis literature provides, in a sense, a philosophical basis to the theoretical paradox popularized by W. K. Wimsatt (1907-1975), which contends that literature is a sort of 'concrete universal.'"
The following figure has often been
offered in this journal as a symbol of Apollo—
Arguments that it is, rather, a symbol of Christ
may be left to the Society of Jesus.
One possible approach—
Urbanczyk's review says that
"Christianity offers the critic
a privileged ontological window…."
"The world was warm and white when I was born:
Beyond the windowpane the world was white,
A glaring whiteness in a leaded frame,
Yet warm as in the hearth and heart of light."
Comments Off on Sunday School
Monday, June 7, 2010
Inspirational Combinatorics
According to the Mathematical Association of America this morning, one purpose of the upcoming June/July issue of the Notices of the American Mathematical Society is
"…to stress the inspirational role of combinatorics…."
Here is another contribution along those lines—
Eidetic Variation
from page 244 of
From Combinatorics to Philosophy: The Legacy of G.-C. Rota,
hardcover, published by Springer on August 4, 2009
(Edited by Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, and Fabrizio Palombi)
"Rota's Philosophical Insights," by Massimo Mugnai—
"… In other words, 'objectivism' is the attitude [that tries] to render a particular aspect absolute and dominant over the others; it is a kind of narrow-mindedness attempting to reduce to only one the multiple layers which constitute what we call 'reality.' According to Rota, this narrow-mindedness limits in an essential way even of [sic ] the most basic facts of our cognitive activity, as, for example, the understanding of a simple declarative sentence: 'So objectivism is the error we [make when we] persist in believing that we can understand what a declarative sentence means without a possible thematization of this declarative sentence in one of [an] endless variety of possible contexts' (Rota, 1991*, p. 155). Rota here implicitly refers to what, amongst phenomenologists is known as eidetic variation, i.e. the change of perspective, imposed by experience or performed voluntarily, from which to look at things, facts or sentences of the world. A typical example, proposed by Heidegger, in Sein und Zeit (1927) and repeated many times by Rota, is that of the hammer."
* Rota, G.-C. (1991), The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT, Cambridge, MA, MIT Mathematics Department
The example of the hammer appears also on yesterday's online New York Times front page—
Related material:
From The Blackwell Dictionary of Western Philosophy—
Eidetic variation — an alternative expression for eidetic reduction
Husserl's term for an intuitive act toward an essence or universal, in contrast to an empirical intuition or perception. He also called this act an essential intuition, eidetic intuition, or eidetic variation. In Greek, eideo means “to see” and what is seen is an eidos (Platonic Form), that is, the common characteristic of a number of entities or regularities in experience. For Plato, eidos means what is seen by the eye of the soul and is identical with essence. Husserl also called this act “ideation,” for ideo is synonymous with eideo and also means “to see” in Greek. Correspondingly, idea is identical to eidos.
An example of eidos— Plato's diamond (from the Meno )—
For examples of variation of this eidos, see the diamond theorem.
See also Blockheads (8/22/08).
Related poetic remarks— The Trials of Device.
Comments Off on Inspirational Combinatorics
Thursday, January 7, 2010
Lesson No. One
| “Zhu Xi maintained that all things are brought into being by the union of two universal aspects of reality: qi, sometimes translated as vital (or physical, material) force; and li, sometimes translated as rational principle (or law).” —Wikipedia |
| “Drop off the key, Lee” — Paul Simon
Reference frame (Click for details.) According to Chu Hsi [Zhu Xi],
“Li” is “the principle or coherence or order or pattern underlying the cosmos.” – Smith, Bol, Adler, and Wyatt, Sung Dynasty Uses of the I Ching, Related material: Dynasty and |
Comments Off on Lesson No. One
Wednesday, January 6, 2010
Epiphany Revisited
|
January 06, 2007 Picture of Nothing
“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….” Related material: The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.” — Tom Wolfe, The Painted Word “Concept (scholastics’ verbum mentis)– theological analogy of Son’s procession as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957, second printing 1963, page 162
“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”
— Jim Holt, Big-Bang Theology, from Slate‘s “High Concept” department  “Bang.” “…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….” For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.
|
Some Context:
See also Nativity.
Comments Off on Epiphany Revisited
Friday, December 11, 2009
Central Ideas
David Brooks today on the historical background of Hanukkah:
"The Greeks had one central idea: their aspirations to create an advanced universal culture. And the Jews had their own central idea: the idea of one true God. The reformers wanted to merge these two ideas."
Related material:
Identity and the rest of March 2008
Monday, December 7, 2009
Something Eternal…
Battleship Gray
“A colour is eternal.
It haunts time like a spirit.
It comes and it goes.
But where it comes it is the same colour.
It neither survives nor does it live.
It appears when it is wanted.”
– Alfred North Whitehead,
Science and the Modern World, 1925
Another Opening of Another Show
"What we've got now isn't really healthcare reform, it's a reshuffling of the deck chairs on the Titanic as far as our patients are concerned, and we're going to make sure that we … have universal healthcare that is truly universal and has eliminated the insurance companies," she told Reuters.
— Deborah Burger, president-elect of the new 150,000-member National Nurses United, comprising union locals from Maine to Hawaii
Comments Off on Something Eternal…
Saturday, November 14, 2009
For St. Lawrence O’Toole’s Day
“We have a need to tell ourselves stories
that explain it all. We use these stories to
supply the metaphysics,* without which
life seems pointless and empty.”
— David Brooks, NY Times of Nov. 10
“The story-teller of hell”
— Publisher’s promotional quotation
for The Nick Tosches Reader
* “the metaphysics“– This link leads to a web page at the Archdiocese of Dublin whose relevance to metaphysics is not obvious. Of course, from the point of view popular with viXra authors (see Thursday), everything is related to metaphysics. The link is to a homily that mentions Sr. Joan Chittester, O.S.B. A search on her works at Amazon.com leads to Welcome to the Wisdom of the World And Its Meaning for You: Universal Spiritual Insights Distilled from Five Religious Traditions. The title indicates that despite Chittester’s personal virtues, her book is, unlike the Tosches book above, less than first-rate. Still, a “meaning for you” is, in my case, not lacking. Continuing the search for a Joycean epiphany related to metaphysics, I found that the Chittester book‘s date of publication (by Eerdmans, the Grand Rapids Calvinist publisher) was July 24, 2007. For a metaphysical phrase on that date– “the Platonic ‘source of all images,'” see The Church of St. Frank. For metaphysics and the Church of some other saints, see the essay on the “metaphysics of goodness” linked to on the publication date of Chittester’s book.
Comments Off on For St. Lawrence O’Toole’s Day
Tuesday, November 3, 2009
Summa Mythologica
Book review by Jadran Mimica in Oceania, Vol. 74, 2003:
"In his classic essay of 1955 'The Structural Study of Myth' Levi-Strauss came up with a universal formula of mythopeic dynamics
[fx(a) : fy(b) :: fx(b) : fa-1(y)]
that he called canonical 'for it can represent any mythic transformation'. This formulation received its consummation in the four massive Mythologiques volumes, the last of which crystallises the fundamental dialectics of mythopoeic thought: that there is 'one myth only' and the primal ground of this 'one' is 'nothing'. The elucidation of the generative matrix of the myth-work is thus completed as is the self-totalisation of both the thinker and his object."
So there.
At least one mathematician has claimed that the Levi-Strauss formula makes sense. (Jack Morava, arXiv pdf, 2003.)
I prefer the earlier (1943) remarks of Hermann Hesse on transformations of myth:
"…in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created."
Comments Off on Summa Mythologica